Report on measurements: Pressure and velocity fluctuations around a granular-bed element

Report on measurements:

Pressure and velocity fluctuations

around a granular-bed

element

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reportBas Heflandno. 06-01

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Report submitted to:

Ministry of Transport, Public Worka and Water Menagement Rcad and Hynrr,uli~ Fn2"iI'e8rb,g Division

Contract no. nww-liüO ""nd

Delft Cluster Themc: O::)~stand river

Project: Behaviourcf Granular S'::'UC7'JTCS

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T

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Delft

Faculteit Civiele.Techniek en Geowetenschappen

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Report on measurements:

Pressure and velocity fluctuations

around a granular- bed element

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Bas Hofland report no. 06-01 December 2001

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Report submitted to:

Ministry of Transport, Public Works and Water Management

Road and Hydraulic Engineering Division

Contract no. Dww

-

1700

and Delft Cluster Theme: Coast and river

Project: Behaviour of Granular Structures

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T

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Delft

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Fluid Mechanics Section,Faculty of Civil Engineering and Geosciences,Delft University of Technology,P.O. Box 5048, 2600 GA, The Netherlands. Tel. +3115 2784069; Fax +3115 27 85975;E-mail:b.hofland@citg.tudelft.nl

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Contents

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1 Introduction 1.1 Background 1.2 Outline . 3 3 4

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2 Methodology 2.1 Introduetion . 2.2 Fields of study . 2.2.1 Bed protections . 2.2.2 Shear stress? . 2.2.3 Extreme forces 2.2.4 Movement . . .

2.2

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5 Bed structure .

2.2.6 Large particles 2.2.7 Far field 2.3 Focal points . . . . 5 5 5 5 6 6 7

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3 Experimental set-up 3.1 Flume 3.2 Cube . 3.3 Stones . 3.4 Measurement techniques

3.4.1 Pressure transducers 3.4.2 Laser doppler velocimeter 3.4.3 Pitot tubes . . . . 3.4.4 Orifice plate in the infiowpipe 3.5 Data processing. . . . 11 11 11 13 14 14 14

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4 Measurement results: bed-mounted cube

4.1 Introduetion . 17 17

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4.2

Previous research . . . .

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4

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3

Inflow

.

..

.

....

.

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4.4

Flow field around cube .

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4

.5

1\verage pressure

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4.5.1

Method

·

..

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4.5

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2

Results

· .

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4.

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Pressure fluctuations

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4.

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Correlation

...

.

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4.7.1

Velocity - pressure correlation

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4.7

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2

Pressure- pressure correlation

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5 Measurement results: granular filter 31

5

.1

Introduction. . .

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5.

2

Stone characteristics

.

.

...

...

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2.1

Porosity ...

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2.2

Topography on micro-scale

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.3

Flow field

.

....

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5.

3.1

Shear stress . . .

34

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3.2

Flow characteristics

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5.4

1\verage pressure

.

.

39

5

.5

Pressure fluctuations ....

40

5

.6

Correlation

.

.

.

...

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..

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5.6.1

Velocity - pressure correlation

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0

5

.6.2

Pressure - pressure correlation

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6 Data analysis 49

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1

Introduetion .

...

.

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..

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6.2

Removing spurious pressures

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Further data analysis . . .

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6.3

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Wavelet

·

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...

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6.3.2

Quadrant analysis

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7 Conclusions 52

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.1

Conclusions

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7.2

Outlook

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References 55

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Chapter

1

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Introduction

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B

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Bed protections made of rock are often used to prevent scouring of the underlying bed ma-terial. The stability of stones in such a granular filter is usually evaluated by means of the

non-dimensional bed shear stress or Shields factor. This factor is a measure of the mobility of the bed; if higher than a certain threshold, the bed proteetion is subject to damage. However,

if the flow is not an equilibrium boundary-layer flow, this approach - using only the mean bed shear stress - does not apply. A good example is the flow behind a backward-facing step. At the reattachment point, the mean bed shear stress vanishes. Still a lot of stones are entrained at this location, due to the increased level of turbulence. The present research is aimed at gaining more insight into the interaction between the turbulence structure and the forces on the stones in order to develop new - physically based - design criteria.

This is the third report made during the present research project (Ph.D. research of author), which was initiated in April 2000. Previously a literature survey was conducted, and the suitability of pressure sensors were examined (Hofland, 2000; Hofland, 2001). In order to get a good picture of the flow processes near and in a granular bed, a number of accurate pressure sensors is used together with a Laser Doppler set-up. The pressure sensors give a detailed view of the pressure field. The combination with velocity measurements gives the opportunity to determine the position and kind of turbulence flow structures which create the extreme forces on stones. This report describes the first measurements on actual stone beds. It mainly presents the results. The analysis is still ongoing. Further progress will be reported at a later date.

The research is financially supported by the Road and Hydraulic Engineering Division of the Ministry of Transport, Public Works and Water Management

(DWW),

and Delft Cluster,

where the project is part of the theme 'Coast and River' , under the cluster project 'Behaviour of Granular Structures'.

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3

1.2

Outline

This report is structured as follows:

The second chapter describes the general methodology of the research. It presents the key

questions about the entrainment of stones from granular filters that are as yet unanswered.

Italso describes the aspects that are being regarded in the analysis in this report.

In chapter 3 the measuring set-up is described.

The following two chapters present the experimental results. Chapter4 describes the

pressures on and flow around a solitary bed-mounted cube. This configuration can be regarded

as a model for a very exposed stone, subject to a flow with low turbulence intensities. This

simple configuration (already with a complicated flow-field) can also be calculated relatively

easily. This generates the possibility to obtain even more information about the entire velocity

and pressure field.

Chapter 5 presents the measurements that have been done on the bed-mounted cube as

part of a (one layer thick) granular filter. The protrusion of the cube has been varied in order

to examine the infiuence of the stone position on stability. In chapter 6 some further possible

analyses on the measurements are described.

Finally, in chapter 7 some conclusions are drawn from the results that have been gathered

sofaro 4

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Chapter

2

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Methodology

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2.1

Introduction

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In this ehapter a few aspects of granular filters are diseussed, in addition to a previous lit

e-rat ure survey (Hofland, 2000). The main conceptual problems in deseribing the entrainment and transport of particles from granular beds are presented. The ehapter is concluded with describing the foeal points of the present research.

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2.2

Fields of study

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B

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d prot

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ctions

A technique used for bank and bed proteetion that is used often is the use of a granular filter. Damage to a granular filter oeeurs when the stones are transported. Therefore, research on sediment transport can be used to study the processes that damage granular filters. In terms of sediment transport, a granular bed proteetion can be described as a bed with the following characteristics:

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• Stone with relatively sharp edges

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• Low mobility (for boundary layer flow: Shields factor =T

j

pgt:::.d

<

0.06)

• Negligible viscous effects

(Re.

=

u.d

j

v

>

400)

• Relatively uniform partiele size

• Non-equilibrium flow and transport

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In this list the following parameters are used: the density of water p, the gravitational accele-ration g, the dimensionless effective density of stone under water t:::.=(Pstone - Pwater)

j

Pwater,

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the stone diameter d, the bed friction 'T, the friction velocity u*

=

.../iJP

and the kinematic

viscosity1/.

A number of causal steps can be discerned in the dependance of bed damage on the flow : Flow -t Pressure -t Force -t Entrainment -t Transport -t Damage

Especially the first three steps are examined in the present research. Therefore, the stones do

not have to move yet. The mobility of the bed is low, so only few particles disturb the flow while moving. This means that as long as the Reynolds number is high enough, the flow is

similar to the flow over a bed with a (very) low number of mobile stones.

2.2.2

Shear stress?

For large grain Reynolds-numbers, viscous stresses on a stone are negligible. Therefore the bed shear stress is actually an averaged drag force on the grains. Even if the flow around

grains would be viscous, already one third of the force on a partiele (Stokes drag) is exerted by pressure instead of shear stress (Schlichting, 1968), which indicates that also then a large part of the bed shear stress is not induced by actual shear stress. Still many people use the

notion of a fluctuating shear stress in entrainment modeIs,although it hardly has a physical basis.

Other models use concepts of fluctuating drag and lift as a factor added to the mean

values, assuming that an increased velocity gives an increased value of the lift and drag force

according to the same formulae as the relation between average velocity and pressure. It is also

then not sure whether this is a useful concept. We state that the fluctuating pressures could

weIl be caused by a completely different mechanism than the average ones. One conceptual

model,discussed in Hofland (2001),gave a possibility. Ifa pressure field is created by an eddy

above the bed, and passes the stone without being altered, this causes a fluctuating force.

From this model it can be shown that wavelengths in the pressure-field of ab out 1.5 times

the stone diameter create the largest forces on the stone. Different kinds of turbulence could

even create different entrainment mechanisms. For the equilibrium boundary layer flow,a

few fairly successful attempts havebeen made to couple sediment transport to the bursting process. This has mainly been done for for highly mobile beds and low partic1e Reynolds numbers.

2.2.3

Extreme forces

Werestriet ourselves to flows with low mobility,as in a well-designed bed proteetion not a lot

of stones will be subject to movement. That is why only the extreme values of the forces will

playarole in the entrainment. The distribution of the extreme pressures must therefore be a

keyissue in the analysis. Which could also be linked to the coherent structures, originating

either from the boundary layer or an external souree of turbulence.

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2.2.4 Movement

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In first instanee only the pressures on the bed and the particles are measured. But for the

assessment of bed damage the partiele must also be set in motion. Therefore not only the

amplitude, but also the duration of the force is of importance. This is an effect that favours large eddies as candidates of being most effective in displacing a stone, as they have a longer duration.

Once a partiele starts to move, ot her complicating effects will occur as weIl. The flow will be distorted by the moving stone. The stone must be accelerated together with a volume of surrounding water, the so-called 'added mass'. Further, the water must also flow into the cavity created under the particle. This will be affected by the porosity of the bed.

Once the partiele is taken higher into the flow, the mean flow velocity is increases and could

be sufficiently large to transport the stone. This means that the extreme force does not necessarily have to lift the partiele above the bed, but a smaller distance could suffice.

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2.2.5 Bed structure

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Unlike research on flow over a smooth bed, research on flow over a rough bed is very limited.

One striking lack of information concerns the texture of the roughness of a granular bed. Even in the absence of bed forms as ripples and dunes, the position of the stones varies such

that the roughness is larger than the classical sand roughness according to Nikuradse. This

variability in positions also causes the critical shear stress to be a stochastically distributed value for all exposed particles. McEwan (2001) has shown this in a discrete-partiele model. A similar one was recently developed by the author, see fig. 2.1. Aberle

&

Dittrich

(2001)

showed that, instead of the diameter of the bed material, it is better to use the measured standard deviation of the bed elevations as the roughness height. This standard deviation is determined from a longitudinal transeet of the bed level. They also showed that it can be advantageous to use it in the Shields parameter for rivers with steep slopes. Their modified Shields parameter now becomes:

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T S

'Ij;

= p!:lgdd

where:

'Ij;

=

Shields parameter,

tau

=

bed-shear stress. New in this factor is the parameter

s,

the standard deviation of the bed elevation. This factor will change if the bed texture alters due to a change of hydraulic conditions. This makes it clear that a change in hydraulic conditions can lead to a different bed structure which alt ers the roughness and the critical Shields factor.

The variety of stone positions poses difficulties for the present research. We want to execute measurements on micro-scale, and subsequently develop a conceptual model of the entrainment processes. A lot of different positions will have to be examined. It might even lead to the conclusion several entrainment are processes possible, and that the total transport is a mixture of them.

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Figure 2.1: Simulated bed-structure on micro scale

Porosity of a granular bed causes the bed-friction factor to increase (Zippe & Graf, 1983;

Gupta & Paudyal, 1985). The friction factor also is a function of the Reynolds number

over a porous bed. This must therefore have an influence on the entrainment as weIl. One

advantage is that the porosity is rather constant for the various materials used for granular bed protections.

2.2.6

Large particles

For the angular rock that is used during the present research, the separation points are fixed at the edges, therefore the critical Shields factor is expected to remain constant for flows

where the thickness of the laminar sub layer is negligible compared to the stone diameter (i.e.

partiele Reynolds numbers higher than about 400), even though there is no empirical proof at

the moment. This is the reason that flume experiments can he used to investigate prototype

situations.

The fact that the critical Shields factor is constant for

Re.

>

400 could alter for stones

with smoother edges, as a consequence of the fact that the boundary layer on the stone

hecomes turbulent at very high Reynolds numhers and the flow will separate from the stone

at a different position. This would alter the forces acting on the stone. Wang & Shen (1985)

report a critical Shields factor of 0.24 (four times the standard value of 0.06) for partiele

Reynolds numbers of 106• Their measurements are not well described, however,so this high

value could be caused by various factors.

2.2.7 Far field

Another question is how turbulence generated by an external souree induces entrainment.

Forthis it is necessary to know how turbulence from the far field interacts with the flow and

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h

Top of ths bound

a

ry

la

ye

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1"-.•,•./'-../" ..,,-..,. I r·· .., ,

)

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(

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urbulent e

dy

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\," ~. . !I

Bre

ak

dow

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;f

Bl

\". ",./ '- ,

~

--'

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en

.·

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•

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f

tllle

eddy

'

,

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"

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'..

orso

....

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ne

..

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w~

. q~

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;::h

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-,

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..

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"""w:s;:_,.,,""~,,_""__ .~,, __~.""',. . .

_

.+_

.;:

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f"

,

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. }

Roughness layer

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Figure 2.2: Model of far field eddy deforming near ground in atmospheric boundary layer, according to Carlotti & Runt

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the turbulence near the bed (near field), which is mainly caused by the stones themselves, and is often linked to the 'bursting process'.

Carlotti

&

Hunt give an example of in an interaction like this for the atmospheric boundary layer, which could very weU be applied to bed protections. When a high mean velocity is present, the strong shear stress near the bed distorts the eddies from the outer regions. This causes the eddies to elongate and brake up into smaller eddies (see figure 2.2). This mechanism could generate simultaneously both small eddies, creating large forces, and a large quasi-steady drag force transporting the stones. This concept might for instanee be used to model the influence of the coherent structures in the mixing layer behind a backward facing step or the wake behind a pile on stone stability.

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₂

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₃

_Focal

_po

_i

_n

_ts

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The present report describes measurements of pressures and veloeities in a turbulent open channel flow over a granular bed. The analysis of the measurements focuses on the following items, related to the above:

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• The extreme forces, and their duration

• Length scale of velocity fluctuations that create the largest forces

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• Originof the eddies that create the largest forces

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Two other items that will be addressed during later research are:

• Interaction of far field turbulence and bed (protection)

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• Bed texture

The turbulence originating from the mixing layer behind a backward-facing step was meant

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as an external souree of turbulence. This has not been realised as of yet. It is planned for

the future, and can be done in the present facility. Also bed texture is not addressed in this

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report, except for the fact that the texture of the granular bed used during the experiments

is measured in detail.

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Chapter

3

Experimental

set-up

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_3.1

_Flume

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In figure 3.1 the flume is depicted. Itis 14 m long, 0.5 m wide and 0.6 m high. The water is

led into the flume from ab ove, which meant that a stilling basin had to be construct ed. The

submerged flow into the flume was led underneath a gate. This caused a wavy water surface in the flume. The waves were damped by a floating polystyrene plate near the inflow. The height of the bed was increased by 8 cm, so that at 3 m from the outflow a cavity could be constructed under the measuring section. The cavity was left open under the plane of the bed at the measuring section for the conneetion of the pressure sensors. In the cavity itself one pressure sensor was placed, to measure the spurious pressures generated by environment al vibration of the flume. During measurements the water levelin the outflow section had to be increased in order to reduce the vibrations of the outflow section, caused by the falling water, which disturbed the pressure readings.

The first series of measurements was executed on a smooth bed, made from 3 mm plastic sheets on the raised floor made of cement. For the second series of measurements stones were placed on the floor manually, one layer thick, over a length of 7 m.

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3.2

Cube

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A cubical shape was chosen for the model stone for a number of reasons. It has sharp edges like crushed stone used for bed protections, it is easy to describe, and the flow around it is fairly easy to calculate. The orientation of the cube was such that one face was placed on the bed, and two side faces were parallel to the flow direct ion. The 30 mm high, hollow cube was welded from 1.5 mm thick stainless steel. Small openings were drilled in three adjacent lateral faces. Pressure transducers wereinstalled in these openings, see figure 3.2. They were all mounted in the center in the transverse direct ion. The transducers at the upstream

(Pl)

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7m

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14m 1.64m 1.12m O.34m. 4.23m O.5Om

longitudinal crou sccti.on

Figure3.1: Flume

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Figure 3.2: Cross section of the instrumented cube (model stone) including three pressure

transducers

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and downstream side

(P3)

were placed at 22.5 mm from the bed

(

i

d),

and the top sensor

(P2)

was placed in the middle of the top face of the cube. The pressure transducers were fixed to the model stone with two-component epoxy adhesive. The electrical wiring and the tube connected to the atmosphere (reference pressure) were led through the bottom of the cube to a cavity under the bed. The hole through which the wiring was led, was sealed off as weIl. In this way the inside of the cube remained dry, with an atmospheric reference pressure.

It

was possible to move the cube to different vertical positions, therefore different protru-sions could be tested.

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3.3

Stones

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The stones were standard stones used in the Fluid Mechanics Laboratory. According to the classification of the CUR-manual (CUR, 1995) the stones can be classified as being shaped irregularly (IR). The roundness of the stones is low,i.e. the edges are sharp. The rocks were sieved with two sieves with openings of respectively 2.5 cm and and 4 cm, in order to get a shortest axis (which point upwards) of about 3 cm length, corresponding to the size of the cube. (The sieve diameter is a limit for the intermediate axis, which is smaller than the axis pointing in the vertical direction, when the stone is positioned on a flat plate.) Extremely elongated or tabular stones were removed. The stones were placed one by one on the floor,

without the use of adhesives. The flow velocity was too small to displace them.

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Characteristic Quantity Full scale 3450 N/m2 Response time 1 ms Repeatability and hysteresis 0.15

%

Linearity 0.25

%

Dimensions housing 6mm x 9mm x 7mm

Material housing Polyetherimide

Dimensions diaphragm 2mmx2mm

Table 3.1: Characteristics of pressure sensors

3.4

Measurement techniques

3.

4.1

Pressure transducers

Miniature, low-range, piezo-resistive pressure transducers are used to capture the fluctua

-ting small scale preesure fluctuations. They are manufactured by Honeywell, type number

24PCEFA1D. The characteristics of the transducer are given in table 3.1. For our purpose

the pressure transducer had to be instalied in waterproof housings. When this was done,

they proved to be rather robust. The transducers need a 10 V direct current amplifier. At

the time of execution of the measurements only four of the required DC amplifiers were avai

-labie. One signal was necessary to measure the environmental vibrations which have to be

filtered from the signal. This left only three pressure signals that could be used for measur

e-ments. The transducers have a rather large temperature-dependence, but as the température

in the flume is very nearly constant (within 0.1°C), this will not be of very much influence.

The transducers can be used with a constant current excitation, reducing the température

dependency.

3.4

.2

Laser doppier velocimeter

A 'Laser Doppier Flow Meter' made by WL

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Delft Hydraulics was applied for the velocity

measurements. The laser has a power of 4 mW. It uses the forward-scat ter, reference-beam

rnethod. The original front lens (600 rnrn) was replaced by a lens with a 300 rnrn focal

distance. This decreased the size of the rneasuring volume to 5 mm long

by

0.5 mm wide.

For yet unknown reasons the LDV could work properly only when rotated over a 90° angle,

illustrated in figure 3.3. Therefore the reference beams were not parallel to the bed, causing

the minimum distance from the bed at which we could measure to be 1.5 cm. The

bed-mounted cube measurements could be done this way. However, it meant that the reference

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reference b

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Figure 3.3: Laser Doppler set-up

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beams were blocked for the measurements with the granular filter. Therefore, in order to get a good signal, a lens with a 400 mm focal length was used. This increased the measuring volume to 10 mm by 1 mmo As the traversing system of the LDV system can only traverse in the vertical direction, the cube was moved horizontally in order to allow for measurements

at different longitudinal positions with respect to the cube. In order to realise the horizontal

movement, the end of the supporting plate of the cube was cut off,so it could be moved.

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3.4.3

Pitot tubes

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At three positions in the flume a Pitot tube was installed in order to measure the water level (see the small, bottom-up triangles in figure 3.1). Only the (hydro-) statie pressure was measured by the tubes. The tubes were connected to gauge glasses installed adjacent to the flume. In these columns, which were not affected by water level fluctuations of short duration,

the water height was measured. It took a long time (about 1 hour) for the water level in the columns to reach the finallevel, the reproducibility of a measurement was within 0.2 mmo The accuracy of one water level measurement is estimated at 0.5 mmo

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3.4.4

Orifice plate in the inflow pipe

An orifice plate was installed in the inflow pipe. The pressure difference over this plate could be measured by a manometer. The pressure difference can be translated into a discharge. Due to the pressure fluctuations, the manometer had a 1 cm water column accuracy. This means an accuracy of approximately 0.71/s.

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3.5

Data processing

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The data recordings were made using the DASYLAB data-acquisition software package. Six

channels were used; four for the pressure signals, and two for the velocity signals. For most

measurements the sampling rate was 50 Hz,except when calibrations were made. Then a much

lower rate was used. MAT LAB was used for the final processing, i.e. making correlations,

spectra, histograms and time-trace plots, .

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Chapter

4

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Measurement

results: bed-mounted

cube

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4

.1

Int

r

oduc

t

ion

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The solitary bed-mounted cube in a boundary layer flow is a flow configuration that has

often been measured and modelled. Therefore it is a good benchmark case. It is also a good starting point for examining stone stability. It represents a very simple model for an exposed stone. Because the surrounding bed is flat the turbulence intensity is relatively low, giving a good reference case for the mean pressures.

First some results from previous research is used to show the general flow field around a bed-mounted cube. After that the measurement results are present ed. First the velocity measurements are described. After that the mean and fluctuating pressures are described. The correlation of both pressure to pressure and velocity to pressure are described last.

Several discharges and water depths were used during the measurements. The dis charge was varied in order to create a range of Reynolds numbers so that it cou1d be seen whether the flow characteristics scaled with the flow velocity. The water height was varied to see whether a waves influenced the bed-pressures. This influence is at least limited to fluctuating pressures with frequencies under 3 Hz (Hofland, 2001). Table 4.1 gives the flow parameters of the various experiments with a bed-mounted cube, and the analyses that were executed with the measurements.

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_4.2

_Pr

_e

_{vious research}

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A lot of calculations and measurements have been executed on the configuration of a surface mounted cube in a turbu1ent stream before the present research. This section is not meant to given oversight of all this research, but rat her to give a general picture of the flow around

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Q h analysis

(m3/s) (m)

0.0435 0.175 velocity above cube and correlation and flow without cube 0.0625 0.225 several verticals for mean flow

and flow without cube 0.035 0.040 0.050 0.060 0.15 sealing and inflow(Q=0.05 m3/s)

0.030 0.040 0.050 0.060 0.15 mean pressure

Table 4.1: Flow parameters of experiments with bed-mounted cube

Figure 4.1: Impression of flow field around bed-mounted cube (Hussein & Martinuzzi, 1996)

a bed-mounted cube. The configuration that was used in the research found, was a

bed-mounted cube in a closed channel, with a wall to wall distance of 2 times the side length of

the cube. This is different than the present open channel flowwith a depth of about 5 times

the sidelength of the cube. It isnot expected that the flow field around the cube is much

different for the two cases,because the flow field is determined locally by the cube.

Figure 4.1 is a sketch of the average flow field from Hussein & Martinuzzi (1996). A horseshoe-vortex is present near the bed. The flow over the cube separates at the trailing edgesof the cube and causes an eddy on top of the cube, and two small eddies at bath sides

of the cube. behind the cube a large eddy is present, whichis shaped like a half torus. The

LDV measurements of the same author of the vertical symmetry plane are shown in figure

4.2. The horseshoe vortex, the top eddy, and the eddy at the back of the cube are clearly

visible.

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y/

H

S.6 . L4 L~

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Figure 4.2: Flow around bed-mounted cube, measured by (Hussein&Martinuzzi, 1996)

4.3

Inftow

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In order to measure the undisturbed flow in front of the cube, the eube was removed, as the LDV set-up eould not easily be moved. The open-channel boundary layer flow measured at this loeation was assumed to be the same as the flow a few decimeter in front of it,so ean be regarded as the 'inflow' in front of the eube.

In figure 4.3 the profiles of the mean and standard deviation of the two velocity components

without the eube present are depicted. Three points ean be seen to deviate from the trend (numbers 4-6 from the top). At the time of measurement of this profile the 300 mm lens was still used (see section 3.4.2). The profile had to be measured close to the bed, so the laser set-up had to be oriented straight-up, whieh deteriorated the signal. Beeause there was not enough signal, another tracker was used in combination with the WLIDelft Hydraulies Laser Doppler set-up. The 3 deviating points were measured with the original tracker, because the quality of the signal at those positions was better.

The points do not eoineide precisely with the trend of the other measurements, therefore the velocity profiles eannot be trusted in detail. The inflow sti11looks reasonably good. The mean streamwise velocity has the expeeted log-profile and the turbulenee intensities are of the expeeted order of magnitude (u' ~ 0.1U, and v' ~ 0.5u') with a maximum near the bed.

It

is difficult, however, to determine the bed shear stress from these profiles, as the best way of doing it, using the Reynolds-stress profile, should mainly be based on the three dubious points.

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200 150

1

';.100 50

.

....... ,-, . ,

.'

o~--~----~

t

~

"'

----~--~----~

o

200 150

1

';.100 50 """'" 0.2 0.4 0.6 U(mis) 0.8 :.

o~--~----~--'~

'

~----~--~

o

:

.

0.1

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0.02 0.04 0.06 0.08 u'(mis) 200 ",' " 150

1

';.100 50 1

.

:

o

~

----

--

----~---~

-0.5 0 V(mis) 0.5 0.1

Figure 4.3: Profiles of mean velocity and turbulence intensities for the undisturbed flow

200 " 0••••••••••••••• ',' ••• ", 150

]

;>-.100 " .: ', . 50 ;

o

0.02 0.04 0.06 0.08 v'(mis) 20

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-

ê

50

I

~ 30

_c

ctS ... CJ)

=e:

_N

20

-I

I

"0 Q) ..c

40

E

o

,._

-I

10~~--~----~----~--~----~--~~

-1

0

o

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Figure 4.4: Mean flow field around eube in the vertical symmetry plane

4.4

Flow

field

around cube

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The measurements with the eube were exeeuted with the rotated set-up whieh gave better results, but points eould only be obtained at positions higher than 2 cm above the bed. Several velocity verticals around the cube have been measured by LDV. Each point was measured for 5 minutes with a 500 Hz sampling rate. In figure 4.4 the vertical flow field in the longitudinal direetion in the center plane of the cube is shown. This is the profile that was measured in greater detail. The eddy on top and the eddy behind the cube are clearly visible (compare to figure 4.2). The veloeity has a maximum at 1 cm above the cube.

The vertical profiles of the mean and standard deviation of the velocity are depicted in figure 4.5. It ean be seen that the fluetuating veloeities near the cube are increased a lot relative to the inflow (fig. 4.3), although the fluetuations in the main flow are still the same. This means that the eddy above the cube is very unstable. Turbulence is generated in the mixing layer between the top eddy and the main flow. The mean flow just above the cube has inereased eonsiderably in comparison to the log-profile, due to the stagnation of the flow caused by the eube.

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21

~ 160 E

.s

140 ] 120 ~ 100 ~ 80 c: nI .]i 60 "0 N 40 20

~----~---~----~----~

-1 , ... ~ . ._... .'

.

.

.',:

.\

..

--0.5

o

0.5 U(mis) ~ 160 E

.s

140 ] 120 ~ 100 Ol ₈₀ u c: nI 60 ~ "0 N 40 20 0 0.05 0.1 0.15 u'(mis)

'

.

_,

'

-: ._. ...•.'. ... - .:~ejI. -.: : ..

.

.

0.2 0.25 40 ...~ 160 ~. 140 • 120 !. 100 .. 80 . ., 60 ·A.

.

j

(

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-0.5

o

V(mis) 20

~----~---~----~----~

-1

I

120 160 . . 140 ..' . 100 . 80 ,. . . 60 40 ....\....:,-v Ó:« Ó: : ,. 0.5

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0.05 0.1 0.15 v' (mis) 20L_ __ ~ ~ __ ~ ~ __ ~ 0.25

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Figure 4.5: Flow characteristics above cube

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-

I'

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>-

'-' $.;

5

Q) <:,) ::l

I

'"0 u:

§

0

.b

I

b.O ~

...

'"0

_-5

cIj Q) $.;

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50

-10~--~----~--~~--~----~----~--~

o

100

150

200

350

t

(8)

250

300

Figure 4.6: Pressure reading while flow in flume is switched of, meant for determination of mean pressure. Flow was stopped at 90 s, water level was at final height at 150 s

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5

A

ve

r

a

g

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pr

ess

u

re

4.5

.

1

Meth

od

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Î

The average pressure had to be determined in an unusual way, because the calibrated range of the pressure sensors was very small compared to the full hydrostatic pressure. Due to the fact that the offset of the pressure transducers was subject to changes in time, it was also not safe to use a measurement that took a great length of time. Therefore it was not possible to calibrate the transducers for a different range.

The following method was used. The flow in the flume was stopped in such a way that the surface level of the stagnant water was approximately the same as the water level in the situation with flow. The water level before and after stopping was measured, and the pressure was recorded at a low sampling rate. A typical pressure recording can be seen in figure 4.6. Because the dis charge was shut off quite abruptly, a standing wave was created in the flume, which can clearly be seen in the figure from 150 s onwards. Because of the varying pressure, and the fact that after the change in mean pressure the offset can change a little, a linear trend line of the water level was based on ten periods of the water level variations, and extrapolated back to the point in time flow in the flume was stopped. The pressure difference between the situations with flow and stagnant water, !::.p, was regarded as the pressure obtained from this

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400

300

200

100

-C\J E .._

0

z

₀ a. 0

-100

-200

0 p1 6. p2

-300

₀ p3 quadratic fits

-400

0.4

0.5

0.6

u (mis) 0.7 0.8

Figure 4.7: Measured mean pressures at the pressure sensors, including fits ofaü2

extrapolation minus the average pressure during flow. The water level was measured as well inthe cases with and without flow. This could be done after the standing wave had subdued sufficiently. The dynamic pressure at the time of measurement, Pd, could then be determined easily by Pd

=

pgb.h - b.p. Here b.h is the measured change in water level. In figure 4.6 it can clearly be seen that the pressure measured by sensor 1 (the upstream sensor) remains much lower than the other two after stopping the flow,showing that the pressure there was higher during flow.

4

.5.2

Results

The mean pressures on the three sides of the cube are presented in figure 4.7. The accuracy of the mean pressure is not very high, due to the way it was determined. The errors of two water level rneasurernents and two average pressure readings are incorporated. Still it can be seen that the pressures on all three sides of the cube seem to have aquadratic dependency on the bulk mean flow velocity (ü

= ~).

This means that the Reynolds number during the measurements was high enough. Viscous stresses can be neglected. The pressure distribution isas expected: stagnation pressure at the front, lift at the top, and a negative pressure with a lower absolute value than the top pressure in the wake. The stagnation pressure is about

24

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half the value of pü2, which is plausible as the flow velocity at the cube is less than the cross-sectional average velocity.

4

.6

Pressure fluctuations

The fluctuating pressures were measured at four different mean velocities. The histograms and power density spectra of the pressures on the three sensors are plotted in figures 4.8 and 4.9. Both figures make it clear that the fluctuating pressures scale very weIl with the bulk mean velocity. The histograms and spectra of the different bulk mean veloeities collapse when normalised by the this velocity. This gives confidence in the measurements, and moreover it

indicates that the Reynolds number is high enough.

The histograms in fig. 4.8 are based on 15 minutes of pressure readings, allowing to obtain a detailed impression of the tails of the distribution. The probability axis plotted is logarithmic. It can be seen that -although the values around the mean are very similar for the top sensor and the upstream sensor- the extreme values for the top sensor are much larger, indicating the fact that the lift forces are fluctuating more. This comparison is not entirely correct, as the high-frequency fluctuations should be filtered out. The corresponding small-scale pressures do not amount to a large force. It can also be seen that the histogram for the pressures at the downstream side are skewed.The magnitudes of the negative extreme pressures are larger than those of the positive ones, which indicates that the extreme values of the destabilising pressures are higher than expected from the standard deviation of the downstream pressure alone.

In the spectrums of the fluctuating pressures (figure 4.9) it can be seen that there is a bump in the spectrum for the top sensor around

f /

ü = 15 m-1. This may be caused by

turbulence eddies in the mixing layer above the cube. At this position we also saw a high turbulence intensity in the velocity measurements. This bump is also seen less pronounced in the spectra of the downstream sensor.

4.7

Correlation

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4.

7.1

V

e

lo

cit

y

-

pr

ess

ure co

r

r

e

la

ti

on

In figures 4.10 and 4.11 the velocity-pressure correlations are depicted. These are meant to study the origin of the pressure fluctuations on the cube. The plots show the coherence function of the velocity at different heights in the flume and the pressures at three positions. The coherence function is the correlation coeflicient per frequency.

It

is defined as:

2

I

G

up

(f)1

2

fUp =

Guu(f)Gpp(f)

(4.1)

where

u

is a velocity component at a certain height, p is the pressure on the cube, Gii is a power density spectrum and Gijis a cross power density spectrum. This coherence function

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~

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o Q=3511s,upstream

o

Q=501ls, upstream

o

Q=601ls, upstream [J Q=3511s,top [J Q=501ls, top o Q=601ls, top t> Q=3511s,downstream I> Q=501ls, downstream I> Q=601ls,downstream o0ot>0 ~

B.

-0.4 -0.2

o

0.2

p

'

/ pu

2 0.4 0.6

Figure 4.8: Histogram of pressures on cube, normalised by mean velocity, collapsing for three

velocities.

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o 0 0 0 0 0 0 - upstream sensor

00 .,0' ':": 0: 0:':0 ': 0 __ top sensor

';! ::: :

1:

1

:

:

:; ~

t

0- 0 downstream sensor

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Figure 4.9: Power spectrum of pressures on cube, normalised by mean velocity. The spectra

are collapsed lines for four different velocities: ij=46-80cm/s.

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was determined for several vertical positions at which the velocity was measured. The velocity was measured right above the centre of the cube. The coherence function gives the proportion of pressure fluctuations, linearly correlated to the velocity component, to the total pressure fluctuations, as a function of height and frequency. The phase lag, (J(f) is determined by:

( ) ImCiup(!)

(J!

=

arctan ReCiup(f) (4.2)

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If a certain phase lag is dominant, this will give a constant, or slightly shifting value of the phase lag over height or frequency. Therefore, if the phase-lag is not random (a random phase is indicated by a speekled image) then that is an indication of some correlation between the flow at that height and the pressures on the cube. It can be seen that areas where no correlation is noticeable from the coherence value still show a fixed phase lag, so some coherence must be present there. As the frequency is connected to the size of the eddy by Taylor's 'frozen turbulence' not ion (>.

=

u(y)j

f),

this indicates the size and position of the pressure sources.

It can be seen in the plots that the pressures at the top of the cube are indeed correlated to the velocity in the mixing layer, as the maximum correlation is situated at approximately 1 cm above the cube. Correlation in this region is present for frequencies up to 30 Hz. The highest correlation between velocity and pressure for the higher frequencies is at 10 Hz. This

corresponds to an estimated wavelength of 6.5 cm. This means that, although the pressure sourees are situated in the mixing layer, the length scale is larger than the size of the cube.

This means that the image of small eddies passing the cube, transported by the mean current

cannot be entirely correct. It might be described in a better way by a non-stationary top eddy. The eddy might also be a vortex shed from the cube at a certain frequency. This might very well have to do with a three dimensional flow pattern. Although the Reynolds number based on the cube diameter is already 24000 for the highest flow velocity, it might still be a kind of a Von Kármán instability in the horizontal plane which causes the top eddy to be unsteady. The fact that the length scale is larger than the cube, could mean that a stone can directly create eddies that are able to generate large forces on other stones downstream.

The low frequency correlations are probably caused by free surface waves, as determined in (Hofland, 2001).

4.7.2

Pressure-pressure

correlation

Correlations between the pressures at the various pressure sensors are made in order to see whether some frozen pressure field is convected over the cube, as assumed in the model described in (Hofland, 2001). The measured pressure-pressure cross correlations are presented in the next chapter (figure 5.13). There they are compared to the pressure-pressure correlation of the cube as part of the granular filter.

coherence (-)

o

0.1 0.2 0.3 0.4 0.5

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:

1

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-50 0 50 pI pI 100 100 80 80 ;-:...

§

60 60 <;» >, 40 40 0 5 10 15 20 25 0 5 10 15 20 25 p2 p2 100 80 80 ;-:...

§

60 60 '-' >, 40 40 0 5 10 15 20 25 0 5 10 15 20 25 p3

o

5 10 15 [(Hz) 20 25 p3

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5 10 15 f(Hz) 20 25

Figure 4.10: Coherence and phase-lag between horizontal velocity component and pressures on bed-mounted cube, as a function of frequency and the height at which velocity was measured

pI pI 100 80 ,.-..

ê

₆₀ '-'

»

40 0 5 10 15 20 25 0 5 10 15 20 25 p2 p2 100 100 80 80 Ä

§

<;» ₆₀ 60

»

40 40 0 5 10 15 20 25 0 5 10 15 20 25 p3 100 80 r=;

ê

'-' ₆₀

»

29

40 0 5 10 15 20 25 0 [(Hz)

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coherence (-)

o

0.1 0.2 0.3 0.4

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0.5 -50

o

50 p3 5 10 15 [(Hz) 20 25

Figure 4.11: Coherence and phase-lag between vertical velocity component and pressures on bed-mounted cube,

as

a function of frequency and the height at which velocity was measured

As the correlation for the lower frequencies is probably largely determined by the waves, the correlation was also determined for two high pass filtered signals, from which the energy content under 3 Hz was largely removed. Both signals are presented in figure 5.13.

30

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Chapter 5

Measurement

results:

granular filter

5

.

1

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ntr

oduction

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For the second series of experiments, a layer of stones is placed in the fiume. Every stone is placed on the floor, so the layer of stone is about one stone diameter high. Even if the same discharge and water level are applied, the flow situation is very different to the flow over the smooth bed case. First of all, the position of the cube (model stone) is different. It is shielded directly by other stones, causing the mean pressures to be less. Second, the mean velocity at the position of the stone is much less, as the granular bed increases the bed friction, yielding a stilllower mean pressure. Third, the turbulence

intensity is higher, which is caused by the

eddies separating from the stones. This eaus es the mean velocity to be even less, relative to the fluctuating velocity. Also the structure of the turbulence itself has changed. For instanee, the longitudinal fluctuations

(u~ms)

on a rough bed are less than those on a smooth bed (Nezu & Nakagawa, 1993).

The water levels were set 3 cm higher than the first series of measurements, as that is about height that is occupied by the stones. Several parameters that describe the filter layer were measured.

The same measurements were done for two protrusions, p,of the cube. The first series of measurements was undertaken with the cube situated at the bed

(p

=

0 cm, see figure 3.2). The second series was undertaken on the cube being raised by one cm

(p

=

1 cm). First the measurements of the stone characteristics are described. After that the results of the flow measurements are described, with an emphasis on the determination of the bed shear stress. After that the measured mean and fluctuating pressures are present ed. Finally the correlations between velocity and pressure are discussed.

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31

5.5 T..·..··· , 5 4.5 4 Ê3.5 u .._. 3 .&::: 2.5 2 1.5 0.6 0.7 0.8 0.9 1.1 (; (h)

Figure 5.1: Measured porosity in granu1ar bed, line is fitted by hand. Level of cube is shown

forp =0 cm (solid line) and p = 1 cm (dotted line).

5.2

Stone characteristics

A few measurements were done on the stones in order to get a good description of their

placement. The porosity as a function of height was measured first. This can give a good

idea of the width of the interface, and the resistance the fl.owis experiencing from the stones

at a certain height.

Secondly,the detailed topography of the stones surrounding the cube was measured. This

is necessary to see which near-field effects from the bed infl.uencethe velocity and pressure

near the cube.

5.2.1

Porosity

The porosity of the filter layer was measured as a function of height, by increasing the water

level in the fl.umeby a few millimeters at a time, by consecutively adding 20 1 of water. The

porosity cou1d now simply be calcu1ated by eq. 5.1.

~v

A

é(h)

=

Kh - 0

As (5.1)

where

ê(h)

is the porosity of the stone bed as a function of

height, f1V

is

the volume of water

added to the flume, b..h is the corresponding change of the water level, As(= 3.5 m2) is the

area of the fl.ume where the stones are placed, and

Ao(

=

3.5 m2) is the area of the fl.ume

where no stones are placed.

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Figure 5.2: Topography of stones surrounding cube. The cube has a zero protrusion.

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The porosity as a function of height is plotted in figure 5.1. It can be seen that at 3 cm height, the porosity is approximately 90%. This value for the porosity can be used as a definition of the bed level in order to compare the results to other research.

5.2.2

Topography on micro-scale

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The topography around the cube was measured by simple means. The supporting plate of the cube was removed from the flume with the stones surrounding the cube still on it. The plate was placed on a table with a mechanicallevelling rod mounted above it. The supporting plate could be moved in two horizontal directions under the levelling rod. In this way the topography could be measured precisely. Longitudinal sections were measured, with the longitudinal spacing depending on the gradient of the bed-topography (1 to 10 mm), and a transversal spacing of 10 mmo The accuracy of the horizontal position is estimated at 1 mm, and the accuracy of the height at 0.5 mmo Figure 5.2 depiets the micro-scale bed topography around the cube with a zero protrusion. When the protrusion was increased to 1 cm, the surrounding bed was not changed. The measured points are depicted in the figure.

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5.3

Flow field

The velocity was measured one cm upstream of the cube, at various vertical locations. The zero level, y

=

0, was still situated at the same level as the smooth bed case, so at the base of the filter. Because about the first three cm was stone in this case, the water level was situated

33

at about 18 cm. Por the case with the highest flow velocity points over the whole vertical

weremeasured, for both protrusions. The flow configurations ofthe various experiments are listed intable 5.1.

5.3.1

Shear stress

The average shear stress is mainly used in stability formulas, so it is an important quantity to measure. There are many ways todetermine this parameter. All methods have in common

that it is difficult to get a precise determination of its value. The direct measurement is very difficultas the stress is very small. The ways in which it was tried are through:

• the water level slope • the Reynolds stress • a log profile fit

The first way in which it was tried to obtain the bed shear stress is by using the water slope.

The depth and width averaged momentum and continuity balanee equations, for a steady

flowover a horizontal bed, are:

düh =0 dx

dh dhü2 -2

ghdx

+ ~

+

CfU

=

0

By eliminating ~~from equations (5.2) and (5.3) it can be determined that:

(5.2) (5.3)

dh

T

=

CfPiJ?

=

p(ü2 - gh)dx (5.4)

80 because of the horizontal bed, the shear stress is not only determined by the water level slope (- pgh ~~), but isalso dependent on inertia effects(pü2 ~~). All parameters ineq. 5.4are

measured. The water level was measured at three positions. The first position, however,was

very closeto the beginning of the rough layer. Therefore the first water level measurement

is not used, and the gradient of the water level is only determined by two positions near the

cube. The shear stresses found this way are listed in table 5.1.

A problem with this two-dimensional approach is that the sidewall influence should be

taken into account as weIl. As the bed ismuch rougher than the sidewalls, and the sidewall

area is smaller than the bed area, it is expected that the infiuence of the sidewalls ean be

neglected. Another problem is the fact that the discharge, and therefore the mean velocity

inequation 5.4 can not be determined very accurately.

The second way is through the Reynolds stress profile. Por an equilibrium flow,this profile

must be linear from the bed. The only way in which a linear profile could be obtained was 34

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' ,

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Q h dh_dx T u.

Re

(m3

js)

(m)

(-

)

(Njm

2)

(mjs)

(-

) p= Ocm 0.065 0.168 0.012 11.8 0.108 130,000 0.055 0.161 0.007 7.8 0.089 110,000 0.045 0.157 0.004 5.3 0.073 90,000 p = 1 cm 0.065 0.166 0.013 12.1 0.1098 130,000 0.065 0.166 0.013 12.0 0.1096 130,000 0.055 0.159 0.007 7.8 0.0881 110,000 0.045 0.156 0.005 5.6 0.0745 90,000

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Table 5.1: Flow characteristics of experiments with filter. From h 3 cm has already been subtracted as the estimate of the bed level.

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to rotate the coordinate axis at every point in such a way that the vertical velocity was zero, see figure 5.3. This yielded angles which varied from 1.3 degrees at the highest point to 7 degrees at the lowest point. As the lowest points are influenced by the bed, the upper three points (plus zero at the water surface) were used to fit a linear trend, which was extrapolated to an estimated bed level of 30 mmo The results were about 30% lower than the shear stress found by using the water level slope approach. The reasonably large angle over which the coordinate axis had to be rotated indicates that this method is probably not fit for a flow with a horizontal bed, as inertia effects, similar to the extra term in the water level slope method mentioned above, influence the result.

The third method uses the fact that the velocity profile must be logarithmic over part of the water column. The integration constant in the equation for the log profile must be iterated in such a way that the points of the velocity profile form a straight line. Now the slope of the line represents the friction velocity. This fit of the log profile did not work. The log-profile is only valid over the lowest 20% of the water depth, but it is also not valid in the neighbourhood of the roughness elements (up to one diameter). These two areas overlap in the present set-up, so it cannot be expected to be a functioning technique.

It

was possible to fit a nice straight line through the points for the case with a zero protrusion, see fig. 5.4, which yielded a T that was very high compared to the other methods (18

Njm

2 for the

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200~--:---:---~---:--~====~~

o o p=û

cm

p=lcm

00

150

... _.', :_{. .}' .'_. '_. . ~

-100

~ O. ... :

:

:

[)..

:

:

: . o

50

.

'0 0 ... 0 . o a.OO DO: 0 o l:.J 0 o

o

1

2

3

4

5

6

7 UV (m/s)2

Figure 5.3: Reynolds stress profiles for both protrusions, and ü

=

0.81 m/s

highest velocity). The measurements with the cube with a 1 cm protrusion had the same bed topography, but the stagnation of the flow in front of the cube caused the velocity profile

to clearly change near the bed, showing that the method was not fit for determining the

(average) bed shear stress.

5.3.2

Flow characteristics

In figure 5.5 the profiles of the mean velocity and turbulence intencity over the the rough

bed are depicted for a discharge of 0.065 m3/s, and a water dep th of about 15.5 cm (plus 3

cm of stones). Normally the peak value for u' for rough beds is less than for smooth beds,

while v' does not change a lot (Nezu & Nakagawa, 1993), this is not the case if the rough bed

results are compared to the smooth bed results. This might be caused by the two different measuring volumes of the LDV that were used for the different cases, given different results.

For the upper part of the water column the fluctuating pressures are near the semi-empirica!

relations of Nezu & Nakagawa (1993).

36

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,

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1.4r---~----~----~----~----~----'---'

1.2 1

~0.8

'-"

~ 0.6

0.4

0.2

o

_o

p=O cm

_p=l

_cm

o~--~----~----~----~--~~--~----~

-5

-4.5

-4

-3.5

-3

-2.5

log(h-ho (m)

-2

-1.5

Figure 5.4: Semi logarithmic plot of mean velocity, for two protrusions, and ü =0.81

mis.

Line is based on the points with a cross inside.

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150 50 o. ···0· . o lij

~

~

..

I ~ ~:~: I

·

o~---~---~---~

o

150 0.5 1 U(mis) GIJ: 0: ··:·0·· o 50 . 1.5

o~----~---~----~----~

o

0.05 0.1 0.15 u'(mis) 0.2 150 50

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o . ·0· o ti}

..

.

....

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·

o~--~--~----~--~--~--~

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-0.4 -0.2

o

V(mis) 150

a

!

100 ,..I:l 50 0.2 0.4 0.6

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a

o

0.05 0.1 0.15 v'(mis) 0.2

Figure 5.5: Mean velocity and standard deviation of two velocity components, for both pro

-trusions. Solid lines represent semi-empirical relations of(Nezu& Nakagawa, 1993)

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