5_1.pdf

Hydraulics Prof. B.S. Thandaveswara

Indian Institute of Technology Madras

5. Secondary Current and Spiral Flow

The curve of constant velocity for rectangular and triangular cross-section obtained by

Nikuradse are shown in Figures 1 and 2. In all cases the velocities at the corners are

comparatively very large with stems from the fact that in all straight pipes of non-circular

cross-section there exist secondary flows. These are such that the fluid flows towards

the corner along the bisectrix of the angle and then outwards in both directions. The

secondary flows continuously transport momentum from the centre to the corners and

generate high velocities there. Schematic diagrams of secondary flows in triangular and

rectangular pipes are shown in Fig. 3. It is seen that the secondary flow in the

rectangular cross-section which proceeds from the wall inwards in the neighborhood of

the ends of the larger sides and of the middle of the shorter sides creates zones of low

velocity. They appear very clearly in the picture of curves of constant velocity in Fig1.

Such secondary flows come into play also in open channels, as evidenced by the

pattern of curves of constant velocity in Fig. 4. The maximum velocity does not occur

near the free surface but at about one fifth of the depth down of the free surface.



Fig. 1. Curves of constant velocity for pipe of rectangular cross-section,after Nikuradse

Fig. 2. Curves of constant velocity for a pipe of equilateral triangularcross-section after Nikuradse

a bFig. 3. Schematic representation of Secondary flows in pipesof triangular and rectangular (open channel) cross-section



water level

Fig. 4. Curves of constant velocity for a rectangular open channel after Nikuradse

Secondary circulation is that flow wherein the velocity can be resolved into two

components, one in the longitudinal direction of the channel and the other in transverse

to the direction of the channel. The transverse component of the velocity gives rise to

the secondary circulation. It can occur in both straight and curved channels and for

different reasons. Secondary circulation is affected by temperature gradients, sediment,

turbulence, non-uniformity of boundary shear, and the curvature of streamlines.

Secondary circulation has been associated with turbulent flow in prismatic channels

wherein the shear at the boundary is not constant. In straight circular pipes as shear at

the boundary is constant for both laminar and turbulent flow the secondary circulation



has not been observed. When secondary circulation does occur, it seems to take place

in an even number of cells as depicted in Figure 5. The non-uniformity of sediment

across a channel has been associated with secondary circulation.

Fig. 5. Secondary circulation in straight channel

Secondary current is the flow taking place in transverse direction of the main flow. The

secondary currents are of four types viz.

1. The 'weak' secondary currents in straight-non-circular channel sections and in pipes

due to boundary resistance (figure 5).

2. Secondary flow developed due to non-uniform bed configuration as in case of alluvial

channels.

3. The ' strong ' currents caused in bends due to centrifugal force.



SPIRAL FLOW

O

y

OUTSIDE WALL

INSIDE

SECTION ON A-A

ILLUSTRATION OF SECONDARY FLOW AND SPIRAL CURRENTSIN A 90 BEND

4. Secondary currents due to the unsteadiness of the oscillating boundary layer.

The occurrence of the maximum velocity filament in a straight channel just below the

free surface (see figure below) to the findings of secondary current.



Secondarycurrents

Isovels

(a) Open channel (b) Equivalent closed conduit

Comparison of Open Channel Flow with Closed-Conduit Flow The lens shaped figure is drawn such that it is orthogonal to each isovel. It may be notedthat the maximum velocity occurs slightly below the free surface. On the lens shaped lineno velocity gradient exist. The shear on the free surface is negligible and their is no shearresistance to balance the component of the weight of the prism along the main flow direction. The equivalent closed conduit is symmetrical about the central line and theshear stress is distributed along the boundary line.

0.750ySo 0.750ySo0.970ySo

y4y

Side Slope, m: 1 = 1.5 : 1

Tractive force distribution obtained using membrane analogyThis distribution varies depending on the cross section and material

Gibson, explained the origination of the secondary current. Darcy, Cunningham, Sterns,

Moseley, Francis and Wood (Thandaveswara, 1969) recognized the presence of this

secondary current and superposition of the main flow leads to spiral flow. If there is any

slight disturbance in approach flow conditions instead of double spiral, then single spiral

exists. Kennedy and Fulton established that the secondary current has a definite effect

on the frictional resistance of the channel.

The second type of secondary currents were observed by Schlichting, Jacob, Schultz

Grunov. The projection of spheres from the surface is just similar to the spherical sand

particles fixed uniformly over the surface, then this type of secondary current can be

expected when the sand roughness is used.

The flow pattern which exists behind an obstacle placed in the boundary layer near a

wall differs markedly from that behind an obstacle placed in the free stream. This



circumstance emerges clearly from an experiment performed by Schlichting and shown

in figure. The experiment consisted in the measurement of the velocity field behind a

row of spheres placed on a smooth flat surface. The pattern of curves of constant

velocity clearly shows a kind of negative wake effect. The smallest velocities have been

measured in the free gaps in which no spheres are present over the whole length of the

plate; on the other hand, the largest velocities have been measured behind the rows of

spheres where precisely the smaller velocities.

d5d

1

2 3

32

1

10d10d

10d

5d

6.00

5.75

5.50

5.25

5.004.754.504.254.00

V[m/s)

measuringstation

Isovels behind a row of spheres as measured by Schlichting. Secondary flowin the boundary layer is marked behind (1), as calculated by K. Schultz-Grunow.In the neighbourhood of the wall, the velocity behind the spheres is larger thanthat in the gaps. The spheres produce a "negative wake effect" which is explainedby the existence of secondary flow. Diameter of spheres d= 4mm

When the spacing of roughness is close, the wavy water surface will not exist as the

formation of vortices will be confined to roughness elements and forms a pseudo-wall

and does not affect the main flow.



s

y

k

Isolated - roughness flow (k/s) - Form drag dominates

s

The wake and the vortex are dissipated before the next elementis reached. The ratio of (k/s) is a significant parameter forthis type of flow



s s

Wake interference flow (y/s)

j j j

Quasi smooth flow - k/s or j/s becomes significant acts as Pseudo wall

s

y

k

y

k

s s s

j

k is surface roughness heights is the spacing of the elementsj is the groove widthy is the depth of flow

Concept of three basic types of rough surface flow

When the roughness elements are placed closer, the wake and the vortexat each element will interfere with those developed by the following element and results in complex vorticity and turbulent mixing. The height of the roughness is not important, but the spacing becomes an importantparameter. The depth 'y' controls the vertical extent of the surface region ofhigh level turbulence. (y/s) is an important correlating parameter.

Quasi smooth flow is also known as skimming flow. The roughness elementsare so closed placed. The fluid that fills in the groove acts as a pseudo walland hence flow essentially skims the surface of roughness elements. In sucha flow (k/s) or (j/s) play a significant role.

In the following paragraphs 3rd type of secondary current has been discussed briefly.

The third type of secondary currents will come into picture while the fluid flows in a

curved channel. The fluid in a curved channel will be subjected to centrifugal force. Due

to this centrifugal force, a pressure gradient normal to the direction of the main flow is

created. Then the particles near the inside wall are thrown outside and they reach the

outside boundary moving in transverse direction. Thus a sort of centripetal lift will be

created causing the heaving up of the fluid. If the flow is irrotational and the fluid enters

with uniform velocity into bend, then it is analogous to the potential vortex.



VELOCITY DISTRIBUTION IN POTENTIAL FLOWIN A CURVED CHANNEL

r

Vr =CONSTANT

O

rircB

v

r0

But in actual case due to the presence of shear stress at the boundary, the velocity of

main flow decreases abruptly at the boundary setting a velocity gradient in the boundary

layer. It may be observed that the energy in the boundary regions is less than in the

potential zone. It follows that at the outside of the bend the pressure intensity falls away

abruptly towards the wall, unless a secondary flow takes place in the direction of outer

wall. Continuity equation requires an inward flow along the side walls to compensate

since the pressure gradient normal to the wall is exactly opposite to that of potential

motion.

The spiral flow motion induced by the centrifugal force is very pronounced and irregular

in the bend. The complicated pattern of flow is caused by the superposition of

secondary current in the bend over the spiral flow of the approach channel. The spiral

flow of bend begins as a lateral boundary current near the point where the stream line

curvature begins and at the bottom inside corner of the bend.

This type of spiral motion also called helicoidal flow and was recognized by Thomson in

1876 and was demonstrated by him in the laboratory in an 180 circular bend with

rectangular channel section in 1879. This was supported further by Engles, Beyerhams

and others. During 1883 to 1990 several researchers while investigating the flow



characteristics in meanderings observed the action of scouring and deposition in the

river bends.

Several investigators (refer to Thandaveswara's Thesis, 1969) mostly conducted the

experiments in channel whose aspect ratios were of the same order of magnitude. Thus

the mean flow occurring was essentially three dimensional in character.

But Betz, Wilcken, Maccol and Wattendrof conducted experiments in two dimensional

channel (rectangular conduit). Watterdrof showed the potential character of the spiral

flow and drew the following conclusions.

(i). There is only slight increase in channel resistance due to the presence of bends as

indicated in pipe bends.

(ii). The velocity distribution follows free vortex law.

(iii). Rayleigh's stability criterion based on the calculation of mixing length and exchange

factor showed the instability and increased mixing at the outer walls of the curved

channels and decreasing mixing and stability at the inner wall.

(iv). If the depth to breadth ratio is large enough so that the lateral currents occupy only

a relatively small part of the area of the cross-section near the bottom and if form losses

are ignored near the bend, then the bend loss scarcely exists.

5.1 Strength of spiral

The term "Strength of Spiral" is defined as the percentage ratio of the mean kinetic

energy of the lateral motion to the kinetic energy of flow and is denoted by xyS .

( )2

xy2

xym mxy 22

V2g V

S = * 100 = * 100VV

2g

The strength of secondary current can be qualitatively estimated to be proportional to

the extent of distortion of isovels. The concentration of velocity near boundary means

the secondary flow concentration near boundary. This bears the hypothesis that the

mechanism of secondary motion arises out of the boundary shear turbulence.

It may be noted that the approach flow plays an important role and has a direct effect on

the number of spirals, strength of spiral and other characteristics of spiral flow.



Following equations relate the deflection angle 1 along the centre line of bed, geometry

of the channel and the hydraulic properties of flow, in channel bends.

(i) For a smooth rectangular bend

(ii) 1

1

c1 e0.25

e

Prtan =17.4 for 2000 R 45000

R

(iii) For a smooth triangular channel

11

c1 e0.25

e

Prtan =13.4 for 2000 R 15000

R

In general,

1

c1 3 0.25

e

Prtan =K

R

If the channel is wide then

1

0 5

c1 4 0.25

e

yr

tan =KR

.

But Russian authors found that for a rectangular wide channel

1c

ytan =11r

In general for a wide rectangular channel,

( )11 0 ec

btan =K Rr

for smooth flow

1 0c s

b ytan =K r K

for rough flow

a1 0

c

btan =K fr



where f = friction coefficient and "a" is an exponent >1. The last equation can be

expressed in Chezy terms of coefficient 8gC=f

in the form

a

1 02c

8g btan KrC

=

The value of 1tan can be assumed to indicate the strength spiral to some scale.

Reference:

Thandaveswara B.S., "Characteristics of flow around a 90 open channel bend",

M.Sc (Engineering), Department of Civil and Hydraulic Engineering, Indian Institute

of Science, Bangalore, 1969.

5_1.pdf

Documents

curves of constant velocity

weak secondary currents

curve of constant velocity

maximum velocity

rectangular pipes

turbulent flow

triangular crosssection

spiral flow