OPEN CHANNEL FLOW Basic terms in study of open channel flow are: discharge or cross-sectional area A, wetted perimeter O, hydraulic radius R h = A/O and.

OPEN CHANNEL FLOWOPEN CHANNEL FLOWBasic terms in study of open channel flow are: discharge or cross-sectional area A , wetted perimeter O, hydraulic radius Rh = A/O and volume discharge Q.

For the rectangular cross-section: (for “real” river h/B0 ; Rh = h).

Irrigation channels are often built with trapezoidal cross-section. Sewage channels are mostly made with circular cross-section.

h

A hB hR

O h B h B

/2 1 2

Partially regulated channels often occur as composite channels.

Natural river beds are nonuniform. The depth is defined as vertical distance between the free surface and the lowest point on the line of cross section.

If the bed is made of resistant material the profile will remain stable and unchanged over time. Otherwise, erosion and deposition will lead to a meandering process.

OPEN CHANNEL FLOWOPEN CHANNEL FLOW

Nature main river bad

Man-made regulation(inundation + dike)

Classification for open channel flow: uniform and nonuniform

Classification for open channel flow: steady - unsteady


laminar and turbulent

Critical Reynolds number for open channel flow D = 4Rh Rekrit 500

subcritical, critical and supercritical

hh

V4RRe =

υ


OPEN CHANNEL FLOW – resistance and turbulen flowOPEN CHANNEL FLOW – resistance and turbulen flowFor the practical purposes one can adopt the logarithmic velocity profile along the vertical:

It is a non-dimensional equation that relates velocity u in particular point of vertical axes, mean velocity V and “shear velocity” u*.

The channel bed with higher roughness induces more intense turbulence. That will cause a decrease in near-bottom velocities, as well as an increase in free surface velocity.

Using the logarithmic velocity profile, Koriolis kinetic energy coefficient has the value 1,04.

*uu y= 1 + 2,5 1 + ln

V V h

The distribution of shear stresses along the contour of trapezoidal cross-section is given in the figure:

The consequence of non-uniform stress distribution is the onset of „weak“ secondary flow in cross section and shift of maximum velocity position from the surface to the deeper layer.

OPEN CHANNEL FLOW – resistance and turbulen flowOPEN CHANNEL FLOW – resistance and turbulen flow

The first task in engineering practice is to calculate the cross-section average velocity V and volume discharge Q for an arbitrary cross-section A. In the case of uniform flow Chezy equation is commonly used:

C - Chezy roughness coefficientI0 – slope of channel bottom

For the uniform flow condition: I0 = IPL= IEL

IPL – slope of piezometric line (slope of free surface) IEL – slope of energy line IEL

To define Chezy‘s roughness coefficient in practice is frequently used Manning's coefficient of roughness n:

h 0V = C R I

6h

1C = R

n

OPEN CHANNEL FLOW – uniform flowOPEN CHANNEL FLOW – uniform flow

OPEN CHANNEL FLOW – uniform flowOPEN CHANNEL FLOW – uniform flow

For uniform and stationary flow conditions Chezy equation would define the relation Q(h) = V(h)*A(h) or so-called “consumption curve”.

OPEN CHANNEL FLOW – local changes in cross-section geometryOPEN CHANNEL FLOW – local changes in cross-section geometryThe occurrence of “sudden” changes in channel geometry causes gradually changes in the flow geometry. The disturbances of free surface are present “upstream” and “downstream” from the position of change.

Intensified vorticity in the vicinity of geometry disturbance is responsible for the “local” losses of mechanical energy:

V – average velocity in flow cross-section 2

V

Vh = ξ

2g

OPEN CHANNEL FLOW – cross-section specific energyOPEN CHANNEL FLOW – cross-section specific energyThe flow through an arbitrary cross-section can be also divided according to the gravity participation in the overall inertia force.

The appropriate regimes are termed as: subcritical, critical or supercritical. We need to introduce the idea of specific energy E of cross-section that is defined by equation:

In order to yield the extrema of specific energy function E we apply the first derivate:

dA = Bdh Froude number (**2)

2

2

αQE = h +

2gA

2

3

dE αQ dA= 1 -

dh gA dh

2

3

αQB = 1

gA

Fr < 1 subcriticalFr > 1 supercritical Fr = 1 critical

The depth at which Fr = 1 one terms as critical depth hkr and associated cross-section average velocity as critical velocity vkr.

If discharge Q and specific energy E are fixed (Q, E = const.), two solutions for h arise from above equation (h1 and h2).

Those two depths approach each other when decreasing the specific energy E. At minimum specific energy E = Emin only one depth can exist and that is the critical depth hkr (Fr2 = Fr =1)

Critical depth hkr for rectangular cross-section is obtained explicitly:

(q = hkr*Vkr )

q = Q/B /V q h2

2

qE = h +

2gh

23 2 q

h - Eh + = 02g

1/32

kr

qh =

g

min kr kr

1 3E = h + h = h

2 2

OPEN CHANNEL FLOW – cross-section specific energyOPEN CHANNEL FLOW – cross-section specific energy

Specific energy diagram (specific energy curve for cross-section) (specific energy E is the function of depth h under the constant specific discharge q)

h>hkr (Fr < 1) subcritical condition

h<hkr (Fr > 1) supercritical condition

IMPORTANT:One curve is obtained by fixed Q (or q) and varying the channel bed slope I


For any available specific energy E exists the corresponding maximum discharge qmax that is “transported” at critical condition (hkr and Vkr).

Example: the flow over a submerged hump .


Flows in open channels will overcome the “obstacles” (weirs) on “economic” way, achieving the critical depth in the vicinity of the structure crest (highest datum-level).

Flow regime upstream of the obstacle is subcritical, on the weir profile or over the submerged wide weir is critical, and downstream of obstacle depends on channel bottom slope (I0<Ikrit , h>hkr, Fr < 1 subcritical I0>Ikrit , h<hkr, Fr > 1 supercritical).


Let us apply the idea of specific energy to the example of steady flow under the plate with the neglected lines and local losses.

The depths before and after the plate (h1, h2) are related to the same specific discharge q and specific energy E.


Raising the plate above the critical depth hkr results in maximum possible discharge qmax for the available specific energy E.


Flow above the hump with relatively “low” denivelation h (we assume neglected energy losses).

subcritical

supercritical

MIRNO (Fr < 1) SILOVITO (Fr >1)

bottom rise dz/dx > 0bottom lowering dz/dx < 0

dh/dx < 0dh/dx > 0

dh/dx > 0dh/dx < 0

width widening dB/dx > 0constriction dB/dx < 0

dh/dx > 0dh/dx < 0

dh/dx < 0dh/dx > 0


OPEN CHANNEL FLOW – overflow and underflowOPEN CHANNEL FLOW – overflow and underflowFlow over sharp-crested weir

P is the weir height, B is channel width, h0 is incoming depth and V0 is average cross section incoming velocity in subcritical regime of flow. Energy losses are to be neglected.

Above the highest weir datum one would find the critical depth :

hp - overflowing depth (vertical distance from weir crest to the free surface at distance of 4hp from the weir cross section).

2

0kr 0 p

V2 2h = E - P = h +

3 3 2g

Considering the critical condition over the weir crest the overflowing discharge can be calculated applying the equation:

After a few steps of editing the above equation one gets:

CQ - nondimensional discharge coefficient

In case of “high” weir hp/P0 ; V00:

The above equation is also used for the other types of weirs. The value of discharge coefficient CQ is obtained experimentally.

Generally, discharge coefficient CQ depends on:

3/22

3 0kr p

V2Q = qB = gh B = g h + B

3 2g

3/2Q pQ = C 2g h B

3/2

Q

1 2C = = 0,385

32

p

p 0 0 0 00

0 σh

ρ

h V V h V,Fr = , , Re= ,We=

P υghform, roughness

OPEN CHANNEL FLOW – overflow and underflowOPEN CHANNEL FLOW – overflow and underflow

More common used form is the ogee weir or ogee spillway with the rounded crest :


Underflow

At a sufficient upstream distance from the gate (plate) the streamlines are parallel and the pressure distribution along the water column is hydrostatic.


The flow is pronouncedly nonuniform in gate cross-section (streamlines are not parallel).

At a certain downstream distance from the gate appears another cross-section with parallel streamlines, so-called contraction cross-section.

The ratio between the gate opening height “s” and contracted depth h1 is termed as contraction coefficient CC (obtained experimentally ; h1 = CC s).


For the flow under the gate of width B following equality are valid:- contraction cross-section area A1=CC s B- continuity equation Q0=Bh0V0 = Q1= BCC sV1

- specific energy is equal for the both of cross-sections E0 = E1 (energy losses are neglected)

In case of horizontal bottom and great upstream depth s/h0 0 ; Fr0 0 the discharge coefficient reads CQ = 0,611.

2 20 1

0 C

V Vh + = C s +

2g 2g

21 0 C 0V = 2g h - C s + V

C C 00

sQ = C 1 - C 2gh sB

h

Q 0Q = C 2gh sB


The transition from supercritical to subcritical flow regime is related to the hydraulic phenomena called hydraulic jump.

Rapid decrease in average velocity and increase of depth, including the high degree of mechanic energy loss takes place in the hydraulic jump.

Because of energy loss hv , the concept of specific energy is no longer valid and applicable (energy loss in hydraulic jump is not apriori known).

Normal hydraulic jump

OPEN CHANNEL FLOW – hydraulic jumpOPEN CHANNEL FLOW – hydraulic jump

We can use the law of momentum conservation and apply it on the control volume that include normal hydraulic jump.

q =V1h1=V2h2

h1,h2 - first and second conjugate depth - averaged tangential stresses at the bottom Lj - length of hydraulic jump

MemberLj is negligible in comparison to the pressure force.

2 21 2

0 j 2 1

h hρg - ρg - τ L = ρq V - V

2 2

0

Normal hydraulic jump


Applying the momentum and continuity equations one defines the relationship between the conjugate depths h1 i h2:

Fr1 - Froude number in cross section where h1

After the calculation of h2 and V2 , one can find the intensity of energy loss (dissipation) within the hydraulic jump by :

212 1

hh = 1 + 8Fr - 1

2 221 2

hh = 1 + 8Fr - 1

2 1

1

vFr = > 1

gh

32 1

v1 2

h - hh =

4h h


The length of hydraulic jump is to be determined experimentally. For the practical purpose one can use relation Lj 6,1h2.

Increase of Fr1 causes the decrease of h2/h1 and hv/E1.

If downstream normal depth h is greater then second conjugate depth h2, hydraulic jump will occur in so-called submerged form. Conversely, hydraulic jump will occur in so-called thrown form that can threat the bottom stability.


If submerged condition is not assured, one has to carry out the so-called stilling basin.

Stilling basin provides the stabilization (localization) of hydraulic jump within its geometry.

Stilling basin is carried out after the spillway, excavating the cave below the level of natural river bottom.


GROUNDWATER FLOW IN POROUS MEDIA GROUNDWATER FLOW IN POROUS MEDIA Darcy velocity v = Q/A applies primarily in the analysis of groundwater flow (flow in porous underground aquifers).

It relies on the assumption of the continuum (v = Q/A ) where the presence of the solid phase within the flow cross-section A is not taken into account.

“Real” velocity is higher then Darcy velocity, what is especially important in the analysis of pollution transport in aquifers.

Darcy experimental device is used for determine the Darcy filtration coefficient (hydraulic conductivity) k of particular filter material.

It is obtained by measuring the Darcy velocity as v = Q/A and pezometric slope I = h/l (ratio of piezometric drop h on path length l ).

Δhv = k

Δl

QΔlk =

AΔh

GROUNDWATER FLOW IN POROUS MEDIA GROUNDWATER FLOW IN POROUS MEDIA

In general 3D case k is the tensor dependent on geological strata and flowing liquid.

Groundwater flow can be observed as potential flow.


hv k I I

lh h h

v ui v j wk u k v k w kx y z

kh kh khv i j k v grad kh

x y z

kh

pv grad h z

g

; ;

FLOW POTENTIAL

physicaly:

In potential flow the flow field is defined with flow mesh that consists of equipotentials and streamlines.


Hydraulic (Dupuit) flow theory neglects the vertical component of flow velocity (equipotentials are vertical lines).

Vertical component of the flow can not be ignored in the vicinity of the well (strong deviation from Dipuit assumption).


Aquifers appear in two characteristic forms, confined (pressurized) and unconfined (with the free surface).


q q gradl

hh

q k h kl l

hk

l l

hk

2

2

2

2

2

2

GROUNDWATER FLOW IN POROUS MEDIA GROUNDWATER FLOW IN POROUS MEDIA In the case of unconfined aquifers it is useful to introduce the concept of Girinsky potential ( specific discharge potential).

We analyze only the simpler cases of wells that are constructed continuously from surface up to the impermeable floor.

nonlinear Q - s relation

2 20H - h

Q =π kR

lnr


Unconfined aquifer

linear Q -s relation

0H - hQ =2π k M

Rln

r


Confined aquifer

Applying the Girinsky potential (in case of pumping from an unconfined aquifer) enables the linearization of the problem, because of the linear relationship between pumping rate Q and Girinsky potential drop .

This allows the application of superposition principle in the case of well group in unconfined aquifer.


x

x x x x x

xx

x x

kH kh; ; ;

kh kH; h H ; s H h;

k

Q Rln LINEAR RELATION and Q

r

2 20

0 0 0

2 2200 0

0

2 2

22 2

2

Application of the superposition principle for a group of wells (two or more wells) in the confined or unconfined aquifer enables the calculation of piezometric height (confined aquifers) and free surface (unconfined aquifer) at arbitrary point in the horizontal plane.


1 2

1 2

1 2

1 1

2 20

0

2 2

2

2 2

x

x

m mi

x ii

xx x

Q R Q Rln ln

r r

Q Rln

r

kH kh

1 2

1 2

1 22 2

x

x

s s s

Q R Q Rs ln ln

kM r kM r

1 1 2

m mi

x ii

Q Rs s ln

kM r


Unconfined aquifer

Confined aquifer

The idea of superposition can be used in analyzing the impact of the open watercourses on the current field in the aquifer.

The watercourse is replaced by a fictive recharge-well that has the same intensity and opposite sign (-Q means water inflow). The fictive recharge-well is placed on the opposite side of the watercourse and on the same distance L from the watercourse.


1 2

1 2

2

1

2 1 0

00

2 2

2

2

2

22

x

x

x

Q R Q Rs ln ln

kM r kM r

Q R Rs ln ln

kM r r

Q rs ln

kM r

SNIŽENJE U ZDENCU:

r L r r

Q Ls ln

kM r


Confined aquifer

WATERCOURSE

The idea of superposition can be also used in analyzing the impact of non-permeable vertical boundary (barrier) on the current field in the aquifer.

The vertical barrier is replaced by a fictive extraction-well that has the same intensity and sign (Q means water outflow). The fictive extraction-well is placed on the opposite side of the vertical barrier and on the same distance L from the vertical barrier.


1 2

2

1 2

2 1 0

2

00

2 2

2

2

2 2

x

x

Q R Q Rs ln ln

kM r kM r

Q Rs ln

kM r r

SNIŽENJE U ZDENCU:

r L; r r

Q Rs ln

kM L r


VERTICAL NON-PERMEABLE BARRIER

Confined aquifer

FORCES ON IMMERSED SOLID BODYFORCES ON IMMERSED SOLID BODYA solid body experiences hydrodynamic forces when it moves through fluid at rest (resistance). The same intensity of hydrodynamic force will be present in situation where the body is at rest and fluid flows around it.

Forces and their intensity are directly related to the viscosity.

At “very” low speed, viscosity is the major contributor in the overall resistant force.

At “high” speed, viscosity has a noticeable effect only very close to the solid body contour.

Occurrence of boundary layer separation from the solid body surface (contour) depends on the body form

FORCES ON IMMERSED SOLID BODYFORCES ON IMMERSED SOLID BODYWe analyze the two-dimensional body of arbitrary shape in Cartesian plane.

Body surface area is A, and the infinitesimal surface element dA is defined with an inclination angle of against the positive x axis.

Stresses along the body contour are divided into pressure (normal) and shear (tangential).

Integrating the stresses along x direction gives drag force:

Integrating the stresses along x direction gives hydrodynamic lift force:

X

A A

F -pcos dA sin dA

y

A A

F -psin dA cos dA

FORCES ON IMMERSED SOLID BODYFORCES ON IMMERSED SOLID BODY

Drag force Fx consists of two parts:- form resistance (first member on the right hand side)- friction resistance (second member on the right hand side)

In the case of flow around the body with symmetry contours lift force Fy is equal to 0.

The primary engineering interest is related to the high-Reynolds number flows where conditions in boundary layer are highly dependent on the form of the solid body.

If the pressure gradient is negative along the body contour, (dp/ds < 0), the boundary layer remains “taped” on the contour. The onset of inverse pressure gradient (dp/ds > 0) triggers the boundary layer separation, accompanied with the eddy formation.


Boundary layer separation takes place at the point of separation A.

In the case of oval body forms (e.g.. pier with circular cross section), the separation point may not be fixed in time.

The intensity of drag force FX is highly dependent on the position A. (A “more on the left” downstream region occupied with the large eddies is larger higher intensity of drag force)

In most engineering problems the form resistance FO has a major contribution in drag force Fx.


IMPORTANT: Created eddies extract the mechanical energy from the main stream, so the integral of pressures acting on the body „second“ half is lower then the integral on “first” half (in x-direction). In case of ideal fluid (rotation-free and inviscid) there is no boundary layer. Consequently, separation and eddy production do not exist, and the drag force is zero.For the practical use one has defined the simple equations for the calculation of drag force FX and form resistance FO:

CX , CO – non-dimensional coefficients of drag and form resistance - density of fluidAP - orthogonal projection of body surface area on vertical plane perpendicular to the flow direction (x direction).

2 20 0

X X P O O P X O

V VF = C ρA F = C ρA C = C

2 2


Coefficients CX and CO are drawn out from the experimentally obtained results:

Generally, drag coefficient CX is the function of body form, Reynolds number, roughness and Mach number (Ma - neglected influence in most of the engineering problems, e.g. for Vair < 200km/h).

If the body is short and has sharp edges, the viscous forces have negligible influence (fixed position of separation, Re has no influence on FX and CX so FX FO and CX CO)

OO 2

0P

FC =

VρA

2


If the body has conspicuous extent in the direction of flow (thin horizontal plate) viscous force dominates in drag force (drag force FX friction resistance force FT ):

X

A

F sin dA

2D forms 3D forms


OPEN CHANNEL FLOW Basic terms in study of open channel flow are: discharge or cross-sectional area A, wetted perimeter O, hydraulic radius R h = A/O and.

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