Evaluation of recent desk study results on granular filters

A

Prepared for:

Rijkswaterstaat

Road and Hydraulic Engineering Division

Evaluation of recent desk study results

granular filters

Desk study May 1998

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Evaluation of recent desk study results

granular filters

HenkVerheij

W L I d e l f t h y d r a u l i c s

CLIENT: Rijkswaterstaat

Road and Hydraulic Engineering Division

TITLE: Evaluation of recent desk study results on granular filters

ABSTRACT:

Recently, a number of desk studies have been carried out with respect to the phenomena in and the design of granular filters. These have resulted in some formulas allowing the dimensioning of granular filters. The most important formulas are those developed by Hoffmans based on the shear stress concept of Grass (indicated hereafter as the Hoffmans/Grass-formula), and the so-called Bakker/Konter-formula. The objectives of the study were:

• Evaluation of the Hoffmans/Grass-formula (based on the shear stress concept according to Grass) and the Bakker/Konter-formula, taking into account aspects as: turbulence intensity, gradation of material, erosion rate, relationship with Shields, extrapolation to non-uniform flow conditions, hypothesis about shear stress. • Determination of the prediction potential of both formulas with relevant available experimental data.

The study yields the following main conclusions:

• The prediction potential of the Hoffmans/Grass-formula and the Bakker/Konter-formula does not differ significantly.

• Both formulas may be extended to conditions with high turbulence and local flow conditions; however, they are not validated for it thus far. Taking into account the theoretical basis the Hoffmans/Grass-formula offers the best opportunities for application outside the experimental range for which it is now validated.

It is recommended that specific experiments be carried out to validate the formulas for highly turbulent flow conditions.

REFERENCES: Commissioned by letter AK980442 dated January 30, 1998 (proposal: letter REN695/Q2416/hw dated January, 22 1998)

REV. ORIGINATO

5

v DATE REMARKS REVIEW / APPROVED BY/ A

2 H.J.Verheij 27-5-98 final report G.J.Akkerman

f t

H.J.Opdam

(1

1

1

KEYWORDS CON" fENTS STATUS

filters, criteria, design rules, bed protection, granular material

TEXT PAGES: TABLES: FIGURES: APPENDICES: 19 2 5 1 • PRELIMINARY • DRAFT x FINAL filters, criteria, design rules,

bed protection, granular material

1

I

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Evaluation of recent desk study results on g r u i u b r fitten Q 2 4 I 6 May, 1998

Contents

1 Introduction 1-1

1.1 Reason and objective of the study 1-1

1.2 Approach 1-1

2 Overview of prediction formulas 2—1

2.1 Introduction 2-1 2.2 Design criteria 2-2 2.2.1 Simultaneous criteria 2-2 2.2.2 Other criteria 2-3 2.3 Background 2-4 3 Discussion on parameters 3-1 3.1 Turbulence factor 3-1 3.2 Grading factor 3-1 3.3 Erosion level 3-2 3.4 Base term 3-3

3.5 Vertical damping factor 3-4

3.6 Extrapolation to non-uniform flow conditions 3-4

4 Accuracy of the prediction formulas 4—1

4.1 Coefficients T) and a 4-1

4.2 Prediction potential 4-5

5 Conclusions and recommendations 5-1

6 Literature : 6-1

Appendix A Derivation Hoffmans/Grass-formula

Evaluation of recent desk study results on granular filters Q 2 4 I 6 May. 1998

List of symbols

c [mH/s] Chézy-coefficient Co [-] dampingcoefficient in B/K-formula ck [-] turbulence coefficient c, [-] constant C| ... c4 H coefficient [-] constant in k-E-model d [m] thickness granular layer

D [m] representative measure for strength Db [m] diameter basematerial

Db x [m] diameter base material exceeded by (100-x)% (weight percentages)

Dchar [m] characteristic value strength Df [m] diameter filter material

Dfx [m] diameter filter material exceeded by (100-x)% (weight percentages) Dt [m] diameter toplayer material

DK [m] diameter toplayer material exceeded by (100-x)% (weight

percentages)

Dt50cr [m] critical diameter toplayer material

Do [m] average value of representative value strength D . [m] non-dimensional diameter granular material e [-] parameter in the B/K-formula

g [m/s2] gravity acceleration

h [m] water depth

ko.b [m2/s2] turbulence energy

P [N/m2] local time-averaged pressure

P' [N/m2] pressure fluctuation near the bed

Pf(tf) [-] distribution function load Ps(t») [-] distribution function strength R [m] largest eddy or hydraulic radius

r _[-] discrepancy ratio

To [-] depth-averaged turbulence intensity related to flow velocity T [m3] transport of material

u' [m/s] instantaneous fluctuation of the flow velocity u0 [m/s] depth-averaged flow velocity

Ubase [m/s] flow velocity in a granular layer not influenced by the velocity in the flow

Uf50,c [m/s] critical flow velocity u(z) [m/s] flow velocity at a depth z u. [m/s] bed shear velocity u .c [m/s] critical bed shear velocity

Evaluation of recent desk study results on granular Alters Q 2 4 I 6 May, I99B

x [m] longitudinal coordinate xm [m] prediction potential

z [m] depth from the transition of flow and top layer

a [-] damping coefficient

a,. [-] ratio between standard deviation and average value of strength (grading factor)

0^. b [-] ratio between standard deviation and average value of strength

for base material (grading factor)

, [-] ratio between standard deviation and average value of strength for toplayer material (grading factor)

a,; [-] skewness factor for non-Gaussian distributions of load

a e b [.] skewness factor for non-Gaussian distributions of base material

( [.] skewness factor for non-Gaussian distributions of toplayer material

cto [-] ratio of O"0 and cok (non-uniform flow conditions)

o^1 [-] product of aa, cob and ck (non-uniform flow conditions)

00 [-] ratio between standard deviation and average value of load (turbulence factor)

P [-] constant

Y [-] erosion level

Yb [-] erosion level base material

yt [-] erosion level toplayer material

Ab [-] relative density base material

A, [-] relative density toplayer material e [m2/s3] rate of energy dissipation by turbulence

1 [-] ratio between local relative turbulence intensity near the bed and the

depth-averaged turbulence intensity

r\ [-] damping coefficient

[-] factor between lc.b and turbulent energy in flow direction

X [-] non-dimensional damping factor

p [kg/m3] density

cc [N/m2] standard deviation strength

aD [N/m2] standard deviation representative value strength D0

O"o [N/m2] standard deviation load

x [N/m2] shear stress

Tc [N/m2] mean value strength

TChar [ N / m2] characteristic value load

TG [N/m2] mean value strength according to Grass

To [N/m2] mean value load

tc,k [N/m2] characteristic mean value strength

Evaluation of recent desk stud/ results on granular filters Q 2 4 I 6 May, 1998

^c,k,transition [ N / m2] characteristic mean value strength at the transition

Xo,k [ N / m2] characteristic mean value load

To,k,lransition [ N / m2] characteristic mean value load at the transition

Tbasc [ N / m2] characteristic value at a depth z

[-] Grass parameter H Shields parameter

H'c.G H critical Grass parameter

Yc,G,b H critical Grass parameter for base material VcG.t H critical Grass parameter for toplayer material

Vc.S H critical Shields parameter

Vc,S,b [-1 critical Shields parameter for base material VcS.t

[-]

critical Shields parameter for toplayer material V b E-] Shields parameter for base material

v . H Shields parameter for top layer material [-] constant cok

[-]

constant subscripts: b base material f filter layer t top layer WL I delft hydraulics i v

Evaluation of recent desk study results on granular Titters Q 2 4 I 6 May, 1998

I Introduction

I. I Reason and objective of the study

Recently, a number of desk studies have been carried out with respect to the phenomena in and the design of granular filters. These have resulted in some formulas allowing the dimensioning of granular filters. The most important formulas are those developed by Hoffmans, based on the shear stress concept of Grass (indicated hereafter as the Hoffmans/Grass-formula), and the so-called Bakker/Konter-formula. For the design of granular filters reference is made to CUR (1993) and Bakker et al (1994).

The Road and Hydraulic Engineering Division of Rijkswaterstaat wanted to evaluate the results of the desk studies carried out, in particular the two formulas mentioned above, and commissionned WL|Delft Hydraulics (letter AK980442 dated January, 30 1998) to carry out an evaluation study.

The objective of the study is an evaluation of the results of different desk studies with respect to granular filters. More specifically:

Evaluation of the Hoffmans/Grass-formula (based on the shear stress concept according to Grass) and the Bakker/Konter-formula, taking into account aspects as: turbulence intensity, gradation of material, erosion rate, relationship with Shields, extrapolation to non-uniform flow conditions, hypothesis about shear stress.

Analysis of the prediction potential of the Hoffmans/Grass-formula and the Bakker/Konter-formula using the relevant available experimental data.

The project has been carried out by WL|Delft Hydraulics and Delft Geotechnics with WL|Delft Hydraulics as leading partner. Ir. H.J. Verheij of WL|Delft Hydraulics was in charge of the projectmanagement, and the report, whereas dr. H . den Adel and ir. M . B . de Groot (both of Delft Geotechnics) provided technical contributions. Ir. G.J. Akkerman and prof.dr.ir. L.C. van Rijn were involved for specialized advices and quality assurance respectively.

Dr.ir. G.J.CM. Hoffmans was the representative of the client, i.e. the Road and Hydraulic Engineering Department of Rijkswaterstaat.

1.2 Approach

WL|Delft Hydraulics carried out physical tests in 1990 (van Huijstee and Verheij, 1991), in which the idea of a design formula based on the simultaneous instability of top and filter layer of a granular filter was checked. These tests resulted in the so-called Bakker/Konter-formula

Evaluation of recent desk study results on granular filters Q 2 4 I 6 May. 1998

(CUR, 1993)(Bakker et al, 1994). Meanwhile, the Bakker/Konter-formula has been modified (den Adel, 1995) and the most recent version is published by De Groot et al (de Groot et al,

1997).

Very recently, Hoffmans published in a memo a formula based on the shear stress concept of Grass (Hoffmans, 1997).

Both formulas will be evaluated. Therefore, in Section 2 a short overview will be presented showing how the formulas were obtained. In addition, other formulas are also presented. In Section 3 the influence of aspects such as turbulence intensity, erosion rate, material gradation, relationship with Shields, extrapolation to non-uniform flow conditions, are discussed. In Section 4 the accuracy of the predicted values is compared with experimental data. Finally, conclusions and recommendations are presented in Section 5.

Evaluation of recent desk study results on granular filters Q 2 4 I 6 May. 1998

2 Overview of prediction formulas

2.1 Introduction

The increasing numerical possibilities for the prediction of local flow fields (amongst others K-e-models) and a recent inventory of stability predictors (Akkerman en Verheij, 1998b) open possibilities for the derivation of formulas for dimensioning granular filters which take into account local flow conditions. Studies related to this subject are: stability concepts for toplayers (Akkerman, 1998a)(Akkerman en Verheij, 1998b), sensitivity analysis of filter design (Hauer et al, 1997), new concepts for toplayer stability (on-going research co-operation project between Road and Hydraulic Engineering Division and WL|Delft Hydraulics).

However, at this moment only formulas dealing with overall parameters are available. At the transition of toplayer and flow three parameters characterize the flow: velocity, shear stress, pressure and their gradients (Figure 2.1). For rough boundary layers the shear stress is a schematical translation of the forces between flow and toplayer material. Since these forces are caused mainly by pressure differences around the stones due to the flow. The turbulent character of a flow is represented by the eddies: above the bed the size of the eddies varies between values equal to the waterdepth and that of the bed material; within the bed the size of the eddies is limited to the size of the voids, which is a fraction of the particle diameter of the toplayer.

-d

Figure 2.1 Schematically the vertical distribution of flow velocity, shear stress, shear slress gradient and pressure gradient

Evaluation of recent desk stud/ results on granular filters Q 2 4 I 6 May, 1998

2.2 Design criteria

Present filter design rules are generally based on a geometrically-tight criterion: the pores of the upper of two adjacent layers must be such that the grains of the lower layer

cannot pass through. Very often such criteria result in too conservative filters with two or three filter layers. To overcome this a geometrically open or hydraulic filter criterion can be adopted. This implies a design, in which the hydraulic loads are too small to initiate the movement of base material. Subsequently, the ratio D/j/Dbso may be larger when compared to

the one resulting from a geometrically tight criterion.

2.2.1 Simultaneous criteria

This idea was the starting point for the derivation of criteria on the basis of simultaneous instability of toplayer material and base material which may also be applied under non-uniform flow conditions. An empirical approach combining various relationships resulted in the Bakker/Konter-formula (Bakker et al, 1994). A more recent theoretical approach is used by Hoffmans applying the Grass-concept which even allows for differentiation in the stability criterion for toplayer material and base material (Hoffmans, 1997).

Thus, this evaluation study concentrates its attention on two equations:

• Hoffmans/Grass-formula, H/G-formula for short, (based on the concept according to Grass, 1970):

gpj _ gflj

D, V c . G . A i>50 D ƒ 5 0 r/ + (l + a0y-rf)exp -ad D, ₍₅₀

WC,GAI

(2.1)

The derivation is presented in a note by Hoffmans (Hoffmans, 1997) and reproduced in Appendix A (see equation A.21).

• Bakker/Konter-formula, B/K-formula for short, in the 1997-version (de Groot et al, 1997): £ >/ 1 5 = 2.2 y/bAb R (2.2) 1 + c2 exp - c , D,

150 J

t R v<

vAsoy

C,= 3.3 c2 = 0.8(0.5 to 1.0) c.3 = 0.8 (0.5 to 1.0) C = 0.25 (0.2 to 0.3) WL I delft hydraulics _{2 - 2}

Evaluation of recent desk study reiulo on granular filters Q24I6 May. 1998

The notation has been used according to (CUR, 1993). The subscript t refers to the toplayer; the subscript ƒ is applied for the layer that fulfills the filter function at the considered transition; and the subscript b is applied for the layer below the considered transition. Applying a toplayer directly on base material (one layer system) means that the subscripts t and ƒ refer to the same layer, which is the situation assumed in this report.

The B/K-formula as presented by eq.(2.2) was originally derived within the framework of a CUR committee with the task to write a manual on filter design (CUR, 1993). Based on earlier measurements for the Eastern Scheldt Barrier the idea was proposed to relate the stability of top layer and filter material. Subsequently, the derived relationship was checked in small-scale flume tests at WL|Delft Hydraulics and shown to be correct. However, the number of tests was limited and only uniform flow conditions were investigated (van Huijstee etal, 1991).

Different versions were published with safety coefficients (Bakker et al, 1994) and with a factor for more severe turbulence compared to "normal" turbulence (Verheij, 1993), etcetera.

The parameter R in the B/K-formula is defined as the dimension of the largest eddy instead of the hydraulic radius (de Groot et al, 1997). In situations with steady flow and a developed boundary layer the largest eddy equals the hydraulic radius. In other situations a corresponding length has to be selected. The reason for this is that the hydraulic radius has no physical meaning in situations with non-uniform flow, whereas the largest eddy does have.

Both formulas result in a ratio between characteristic diameters taking into account flow conditions, turbulence and material characteristics, but for the moment they are validated for uniform flow with normal turbulence conditions. However, both can be applied for geometrically-open transitions between two layers under various hydraulic conditions, for both uniform flow as well as for directly downstream of hydraulic structures such as sluices, locks, barriers, piers, etcetera, viz. structures generating a high degree of turbulence. With respect to the H/G-formula the aspect of non-uniformity is dealt with in Section 3.6, and in the B/K-formula it results in a larger diameter D,5o of the toplayer material.

It should be noted that both formulas may be applied i f toplayer material and base material are divided by an extra filter layer. In that case for the thickness d the complete thickness of the filter (toplayer and extra filter layer) must be substituted.

2.2.2 O t h e r criteria

Earlier, in 1979 Stephenson published an equation which relates the ratio of grain diameters to the ratio hydraulic radius - toplayer diameter (Stephenson, 1979):

EL^IJ&LJL (2.3)

A

v A

A

Evaluation of recent desk study results on granular filters Q2416 May, 1998

Assuming an average value of 21 for the ratio R/D, (range 6.7 to 40 corresponding with the available experimental data mentioned in Tables 4.1 and 4.2; van Huijstee et al, 1991) and equal properties for base and toplayer material the result is:

D

^ « 1 6 8 A

De Groot et al (1997) concludes in his study that

(2.4)

r

D A ^ « 8 0 (2.5) 650

has an equal prediction potential as the B/K-formula.

The H/G-formula takes into account the rate of turbulence. For common situations values of Oa = 0.4, Oc = 0.3, y= 0.625, a= 4.2 and 7] = 0.012 (last two values presented by Hoffmans (1997), see for a better fit Section 4.1) can be substituted, resulting in:

A/IS

=

gffi

A 1.25 _{V c . G . A} 650 D / 5° 0.012 +1.238exp -4.2 d

A

50 VcaA (2.6)

r

This formula is very similar to a formula presented earlier by Hoffmans (Hoffmans, 1996a):

D f\ s _ Dns

v A

exp ' d ^

. A50

j

A50

D/so ¥A,

This equation was slightly modified by De Groot et al (1997) into:

Df \ i _ Df \ s vAt

A

5 0

D/50

VA, exp

A

if \ - ^ ( 5 0

J

D, ₁₅₀ <2

A50

A/ 5 0 i / , A ,

Aso

(2.7) (2.8)

The reason for this modification was that according to experimental results of Shimizu et al (1990) the damping factor a should have a maximum value of 2. The value of 0.8 for the coefficient c3 in (2.2) is based on the same idea.

2.3 Background

The shear stress concept of Grass is basically a statistical extension of the design methods for erosion control as introduced by Shields. From a physical point of view, erosion is a matter of

Evaluation of recent desk stud/ results on granular filters Q 2 4 I 6 May, 1998

balancing load and strength. Shields related the shear stress on the bed to a critical size of the bed material. The shear stress is the load, whereas the size of the bed material is a representative measure for the strength of the bed material:

T = pt£ = y/cSApgD (2.9)

This is an oversimplified model of the tricks erosion has deviced for us. In fact the shear stress on the bed is not constant: it varies in time and space. To complicate things a bit more, the bed material isn't homogeneous in size or packing either. When a model for erosion is extended to follow the capricious nature of turbulent flow, the first step is to characterize the shear stress by an average value and a standard deviation cr0 . Similarly, the strength of the

bed can be characterized by an ensemble averaged value of the size of the bed material, D0,

and a standard deviation in the size: ofa. Shields' relation links the time averaged value of the shear stress to the ensemble averaged diameter of the bed material, so:

t0 = pw.2 = y/cSApgD0 (2.10)

Schukking et al (1972) and Breusers et al (1971) have performed tests in which both the shear stress and the critical size of the bed material were measured and have extended the Shields data by introducing a measure of transport of bed material:

1. occasional particle movement at some locations 2. frequent particle movement at some locations 3. frequent particle movement at many locations 4. frequent particle movement at nearly all locations 5. frequent particle movement at all locations 6. permanent particle movement at all locations 7. general transport (initiation of ripples)

For each of these criteria they found curves similar in shape to the Rouse curve which is based on Shields' experimental data, Figure 2.2.

Figure 2.2 Criteria for particle movement (Schukking et al, 1972)

Evaluation of recent desk study results on granular filters Q 2 4 I 6 May, 1998

Grass applies the statistical method of characteristic values for load and strength. The characteristic value for the load is:

Tch*r = T0+r°-0 (2.11)

whereas the characteristic value of the strength is:

Dchar=D0-y°'D (2.12)

Furthermore Grass assumes that the standard deviation in both load and strength, a0 and crD

can be expressed in a fraction of their averaged values:

°~o = a0To a n d ° D = aDDo (2.13)

Finally Grass assumes the same relation between the characteristic shear stress and critical size of the bed material as Shields uses for the averaged values:

^ehar = ¥e,G^PgDchor (2.14)

Substitution of equation (2.13) into equations (2.11) and (2.12) and substitution of the modified equations (2.11) and (2.12) into equation (2.14) and combining with (2.10) leads to the relation between the Grass parameter y/0 and the Shields parameter y/s:

l + a0y

WC,G = ¥C,S- — (2.15)

\ - a j

With ceo = 0.4, a, = 0.3 and y= 0.625 (according to Grass, 1970) follows: y/c.G = 1.54 y/c

c.S

Applying y means that this parameter is the design variable representing the rate of erosion instead of y/s. The parameter r//G becomes a fixed value related to the Rouse curve (criterion 5

according to Schukking et al, 1972; see Figure 2.2). Increasing the value of pleads to less transport.

In this respect, it is important to mention that in eq.(2.1) the stability predictor for the top layer material is based not only on verification tests carried out by Grass, but also shown to be valid for coarse materials up to coarse sand D50 = 2.6 mm (Boutovskaia, 1997).

The y= 0.625 corresponds to the Schukking criterion 5 (Boutovskaia, 1997). However, criterion 5 is often not applied but a lower one for which y= 1.0 resulting in: ^ G = 2.0 y^s

Implicitly, eq.(2.15) which is the basis for the Grass-concept makes already clear that the Shields stability equation for uniform flow conditions is hidden in the H/G-formula. In the Appendix, where the full derivation of the H/G-formula is presented, the introduction in eq.(A.13) and (A. 17) of relations according to Grass between characteristic shear stress and critical material sizes for bed material and top layer material respectively, results in the term

WcG.b/^cG.i- This term can be replaced by \f/c,s,b/Wcs,t because the ratio is equal.

Evaluation of recent desk stud/ results on granular filters Q24I6 May, 1998

Finally, a remark should be made about developments at Delft University studying with the forces on individual elements of the toplayer and base material (Booij, 1998). This micro approach will not result into applicable results within a short period of time, but in time this approach may lead to practical results.

WL I delft hydraulics

f

I

Evaluation of recent desk study results on granular fitters Q24I6 May. 1998

3 Discussion on parameters

In the H/G-formula various parameters appear:

• do turbulence factor • Oc grading factor

• y erosion level, or the level of failure or safety

• 77 base term

• a damping of shear stresses with depth

• extrapolation to non-uniform flows

Each of these aspects will be discussed in this chapter. A recent study carried out within the framework of the cooperation between Delft Hydraulics and the Road and Hydraulic Engineering Department, could be used fruitfully (Akkerman en Verheij, 1998b).

3.1 Turbulence factor

The parameter ceo is a factor which may be used to express the turbulence level. For uniform flow conditions Grass derived from his experiments a value of On = 0.4, which agrees with the results of experiments by others (see amongst others: Compte-Bellot, 1963; de Ruiter,

1982). For higher turbulence levels this value may be chosen higher.

It should be noted that the distribution of turbulence is not a normal distribution; for high turbulence deviations occur. In general the distribution should be characterized also by a skewness factor (%,. Instead of the term (I + o^y) the term (Akkerman en Verheij, 1998b):

should be substitued with ctv= factor for the skewness ( o ; = 3 for a normal distribution). At the moment sufficient information on this point is not available. In order to introduce some skewness an alternative is to assume a log-normal distribution.

3.2 Grading factor

From the construction and economic points of view it may be attractive to apply rather graded material to protect bed material. The distribution of the strength should then be considered. The advantage is that higher loads are allowed or as an alternative the ratio

D/i/Dbso is enlarged. This can be explained as follows.

In the H/G-formula as presented by eq.(2.1) the aspect of graded materials is not taken into account (the factor etc disappears since it was assumed that both toplayer and base material

Evaluation of recent desk stud/ results on granular fitters Q 2 4 I 6 May, 1998

have the same distribution). However, in equation (A. 18) in the Appendix the effect is present in the term:

L L-£ è- (3.1)

in which ac,b is the ratio of standard deviation and average value of the base material related

shear stress, and is the ratio of standard deviation to average value of the toplayer material related shear stress.

Obviously, the denominator of eq.(3.1) decreases i f the parameter aCJ increases and,

subsequently, the ratio Dfn/Dbi0 increases. The inverse effect occurs by increasing the value

of aCib.

In general, the following relationship is valid:

°~c = acTc = acy/cCApgD50 (3.2)

Analogous to the skewness in case of non-Gaussian distributions of the load also for the materials a skewness factor may be introduced (Akkerman en Verheij, 1998b):

~o~ W

From experiments (Grass, 1970) derived an average value of 0.3 for the factor a^,, for uniform material in the range 75 to 210 urn.

For graded materials higher values may be expected. I f sieve curves of the materials are available values may be derived.

Introducing (3.3) into the H/G-formula 2.1 has the advantage that also graded materials may be considered, including different erosion levels for filter layer and base material and different skewnesses.

3.3 Erosion level

The erosion level is indicated by the factor y and can be derived from x0,k = Tc,k resulting in:

y = ^ ^ (3.4)

<J0 + ac

For the extreme y= 0 the stability criterion becomes r0 = Tc.

Another extreme occurs for y= 3: hardly any material will erode, since r0,k = to + 3 a0 and

Tc.k= Tc-3 Cc.

WL I delft hydraulics

Evaluation of recent desk study results on granular filters Q 2 4 I 6 May, 1998

Based on the results of Grass and Boutovskaia a value of 0.625 is recommended for the H/G-formula. This value corresponds to the Schukking criterion 5. For more stability a higher criterion should be selected, for instance Schukking criterion 3. This requires a higher value for y, for instance 1.0.

The selected value influences the relationship between ^ G and ^ s (see 2.15):

Wco = 1-5 Yes for ^ = 0.625

y/co =2.0 yfcj for y= 1.0

As shown by equation (3.3) different erosion levels may also be introduced.

3.4 Base term

The base term represents the quasi-steady groundwater flow driven by the flow above the toplayer. Instantaneous fluctuations are no longer present. The base term is defined as:

those = TJTo (3.5)

From data fitting (see Hoffmans, 1997; or Section 4.1) a value of about 77 = 0.01 follows. With T0= 0.33 p' = 0.33p (uf (Breusers, 1972) and u' = 0,10 m/s near the bed (Verheij,

1997) follows: rbase = 0.033 N/m2.

This value may be compared with a value calculated with a relationship for Tbase derived by

Hoffmans (1996b)(also presented in Van Os, 1998):

Tbase-T0 j (3-6)

C, h

with p= 0.02 and c, = 0.08 (Hoffmans, 1996a).

In the experiments carried out by Verheij (1997) values for Dp0 and h were 15.2 mm and 0.35

m respectively, which results in: Tbaie = 0.031 N/m2. This value is nearly similar to the earlier

value derived with the fitted value of #7 = 0.01.

The experiments showed that low frequency velocity fluctuations do not penetrate into a filter structure, but pressure fluctuations do, although the influence diminishes rapidly with increasing depth.

It is concluded that the base term is very small, but the influence may not be ignored. This depends on the thickness of the filterlayer

Evaluation of recent desk study results on granular filters _Q24\6

May, 1998

3.5 Vertical damping factor

The damping factor a varies: Hauer(1996): Hauer(1997): Hoffmans (1996): Hoffmans (J997): den Adel (1995): a = 3.0 to 3.6 or=2.0to5.0 a = 2.0 a=4.2 a = 0.8

The Hoffmans (1996), the Hauer (1996 and 1997) and the Den Adel (1995) values are based on experimental results of Shimuzu et al (1990), that is to say: 4 experiments. The other value (Hoffmans, 1997) is derived from the experimental data in Van Huijstee et al (1991). In Tables 4.1 and 4.2 the main results of these experiments are summarized and the ratio d/Dpo

is in the range 1.0 to 4.4.

A maximum value of 2.0 assumes that pressure fluctuations will be damped for layers thicker than two times D50. On the basis of this assumption De Groot et al (1997) modified the

Hoffmans equation into equation (2.7).

Using the data as presented by Van Huijstee et al (see Table 4.1) the values of c, and c3 in the

B/K-formula are fitted again resulting in:

cj = 3.0 and c3 - 0.46

More experimental data are needed to determine a more accurate value and its variation.

3.6 Extrapolation to non-uniform flow conditions

Equation (A. 17) in Appendix A represents a general relationship for uniform and non-uniform flow conditions including the general stability equation for a toplayer according to (A.5): D ISO _t0+ r , o -0 D ₆₅₀ 77 ^ o + ( ^ " o + r , o "0- 7 7 r0) e x p -a d ISO J (3.7)

After multiplying both sides with A / / A J O a general design equation will be obtained comparable with the H/G-formula, but for non-uniform flow conditions:

A ,

5

_

A . 5 _lo±r,o-0 A s o A 5 0 , / \ nto+iTo+r^o-r/T^exp -ad L A 5 0 J i - r ^ c . r Vc.GA, (3.8) WL I delft hydraulics 3 - 4

Evaluation of recent desk study results on granular filters Q24I6 May, 1998

Formulas similar to (3.8) may be derived, but related to the local flow velocity near the bed. This requires other stability predictors. In (Akkerman en Verheij, 1998b) the first results are presented of research on this subject.

For conditions with non-Gaussian strength distributions the term as presented in Section 3.2 may be introduced.

In his thesis Hoffmans (1992) derived for a flow downstream of a sill a relationship for a0

consisting of a geometrical term and a base term:

in which

<r0=palaul+pa0ul (3.9)

<** = aaa>bck with aa=^±

For uniform flow conditions the value of cok is approximately 1 / Jc^ = 3.3 where cM = 0.09

(constant in k-E-models). With a0 = 0.4 follows: aa m 0.12.

Values for eob and ck are 0.3 and 0.035 respectively (Hoffmans, 1992), resulting in: a j *

0.0013.

For T0 may be substitued: r0 = pul.

The first term in (3.9) represents the additional turbulence in a non-uniform flow due to the geometrical situation. Eq.(3.9) can also be written as (by substituting aa and aj):

o-0 = paaa)bckul +pa0u2. (3.10)

Introducing u>2 in the first term at the rigth hand side and subsequently replacing pu- by r0

results in:

'coh ul \

{mk u. J

With kob = cokul follows:

(3.12)

Obviously, neglecting the first term which is the situation for uniform flow, results in the earlier presented formula for uniform flow.

Note: For uniform flow conditions kob = rj^^ul) and the values of some of the

parameters read: £ = 1.6, rn = 0.9, cu = 0.09 and r0 = 1.45 g / C2 (see Rodi,1980;

Hoffmans, 1993).

For non-uniform conditions values of £ are smaller than 1.6 and r|t lies in the range of 0.9 to

1.5.

WL I delft hydraulics

Evaluation of recent desk study results on granular filters Q24 16

May, 1998

In a uniform flow CT0 is small compared to x0 whereas with increasing the non-uniformity of

the flow conditions a0 increases as well.

WL I delft hydraulics

Evaluation of recent desk stud/ results on granular filters Q 2 4 I 6 May, 1996

4 Accuracy of the prediction formulas

The prediction potential of both formulas is checked with the available data from (van Huijstee et al 1991). Experiments were selected in which simultaneous instability of both base material and toplayer material was observed. Nine experiments meets this demand (Table 4.1) , in 11 experiments instability of both layers occurred at different levels of loading (Table 4.2) .

Before checking the prediction potential of both formulas the values of two parameters in the Hoffmans/Grass-formula have to be fitted to the data: the base term rj and the damping coefficient a.

4.1 Coefficients T| and a

The H/G-formula has been derived for simultaneous instability of base and toplayer material. Therefore, only the data in Table 4.1 can be used for the determination of the values of 7] and

or.

Before presenting the results, some remarks are made about some other values in the Tables. The presented data differ from those earlier presented by De Groot et al (1997) which is the result of a reassessment using new insights. The differences deal with different values, but also some datasets are moved from one table to another.

The values of y/b and y, are determined with the relationship between ^and D* as presented

by Van Rijn. The values of iip0,c are derived from the original tables and figures and are used

to determine the critical grain size of the toplayer Dpo,c

-The fitting has been carried out assuming do = 0.4 and y = 0.625. -The last value may be somewhat too low, since it proved to be valid for the Schukking criterion 5. As a complicating factor in the experiments criteria were assigned which differ from those of Schukking. The instability criterion 3 in the experiments is comparable to Schukking criterion 4. Subsequently, the value for y should be about 1.0.

The result of the data fitting is:

criterion 5, i.e. y = 0.625: 7/= 0.0069 a n d a = 3 . 7 criterion 3, i.e. y=l.0: TJ= 0.0077 a n d a = 3 . 7

The factor or proves not to be sensitive for the experiments carried out. Note: the value fits in the range as mentioned by Hauer et al (1997): 2.0 < a< 5.0 or Hauer et al (1996): 3.0 < a< 3.6. It does not fit in the idea of a maximum of 2.0 (see Section 3.6)

A 50% higher value of y results into a value of rj only 10% higher.

WL I delft hydraul ics

Evaluation of recent desk study results on granular filters _Q24I6 May, I99B to ii CD t o CM CD ö II T3 CD CO CM t O CM N O ) CD CO CO LO O) | o CO I CD I CO O) O CM O) CO co o> i n CM I T— i n t o 10 CO CO CO O O O Ö O ö r-~ co -er CM o o O O O , o i o CO CM CM O lO lO CM r - T -t o t o co CM O O o i o co co rr o | o ' LU co CM c 0 CD i - y I -CM C M CM r - t-- r» t o U •) l O a c t O

Table 4.1 Data of tests with simultaneous instability of base and toplayer material

WL I delft hydraulics

Evaluation of recent desk stud/ results on granular filters Q 2 4 I 6 May, 1998 CO i i CD O r IT) n o V ** I CM o> O CM to co I M" to If) <o k o r- hol 0. CD Q .

I>5

CO I T— I OJlCO " * CO O) CO CO E 1 " O E l ? to E «) if) to C O | C M | to CM s l o i o CM CM to to I to CO CO CO O O [ O o' ö h~ CM O O O O I O I O CM CM r— to to co L O I T J - I L O O

I °

I C i tn O I O I O co co co E l ö CM r-LO O o | o' | o' CD O lO (o I - h¬ I I CM CM to to O O CD CM I CM CM CM CM CO| T f 00 l O LO CO CO co co CM CM C M | ° -Q I •a CD o CO Ü O < in ö o O \n CM f 5 u 3 ~

£

II Ü O o" II LD O O £ " . e _[|M wi l T 3 "O • o Q) QJ 0 ) ro CO _ro LU I LU j LU J (/) V) co CD CD CD "E CO 17) <D E co (/) CD CO CO CD e w '</> co CD O CD -Q t= CD >> _CÜ Q. O to co L_ O • to co E J5 l i S-CD LU I CD C C O O V (/)

I I

co co i i i i * r -CQ CO

Table 4.2 Data of lests for earlier or later instability of base material than toplayer material

WL I delft hydraul ICS

Evaluation of recent desk study results on granular filters Q24I6 _{May, 1998}

Values for 77 and or are also fitted using all experiments, viz. Tables 4.1 and 4.2 using y = 0.625, leading to:

77= 0.0090 and a=l.\

Clearly it can now be seen that values of both 77 and or increase. This value of a differs significantly from the range of Hauer et al and tests.

Hence, for the analysis of the prediction potential of the H/G-formula and the B/K-formula in section 4.2 the following values will be used:

77= 0.007 and a = 3 . 7

4.2 Prediction potential

In this section values predicted with both formulas will be compared with experimental results as mentioned in Van Huijstee et al (1991). The results are obtained with y= 0.625, 77 = 0.007 and a= 3.7. The predicted results with respect to Dtl5/DbS0 are given in Tabels 4.1

and 4.2 and Figures 4.1 to 4.5.

It should be noted that the values calculated with the B/K-formula are based on the values of the coefficients mentioned in Section 2, which were originally the result of a fit carried out by Den Adel (1995). This implies that the prediction accuracy may be less.

Figures 4.1 to 4.5 clearly indicate that both formulas predict the measured values within a range of 1/3 to 3 times the line of perfect agreement.

Comparison Hoffmans Grass with measurements

• Q572

A Other experiments

O Q 5 7 2 '

O Other experiments *

Df15/Db50 calculated according to Hoffmans Grass, y = 0.625

Figure 4.1 Comparison of measured data and calculated data with the H/G-formula

WL I delft hydraulics

Evaluation of recent desk study results on granular filter» Q 2 4 I 6 Mjy , ,98

Comparison Bakker Konter with measurements

O Q 5 7 2

A Other experiments

O Q572 *

• Other experiments"

Df15/Db50 calculated according to Bakker Konter

Figure 4.2 Comparison of measured data and calculated data with the B/K-formula

Comparison Stephenson with measurements

IQ572 Other experiments 0.572* Other experiments * WL | delft hydraulics 4 - 5

Evaluation of recent desk study results on granular filters Q24I6 May, 1998

Comparison Hoffmans Grass with measurements

• Data from Table 4.1 • Data from Table 4.2

Df15/Db50 calculated according to Hoffmans Grass, y = 0.625

Figure 4.4 Comparison of measured data and calculated data for the H/G-formula with respect to simultaneous and nearly simultaneous instability

Comparison Bakker Konter with measurements

350

• Data from Table 4.1 • Data from Table 4.2

DH 5/Db50 calculated according to Bakker Konter

Figure 4.5 Comparison of measured data and calculated data with the B/K-formula with respect to simultaneous and nearly simultaneous instability

WL | delft hydraulics

Evaluation of recent desk stud/ results on granular fitten Q 2 4 I 6 May, 1998

Also the prediction potential xm ( defined as the ratio of the sum of the calculated values to the

sum of the measured values) is determined. However, a first order moment does not give an indication of the variance of the results, and subsequently a percentage has been calculated representing the number of results within a defined area. The upper and lower limit of this area are determined arbitrarily. The discrepancy ratio r is defined as the ratio of the number of figures within the interval to those outside (see the sketch below).

20

15

0 5 score: 2 / 6 x 1 0 0 = 33%

o Experiments

Line of perfect agreement

Figure 4.6 Definition discrepancy ratio

The results are shown in Table 4.3 for the data of Table 4.1 (on which the fit has been carried out for the H/G-formula) and in Table 4.4 for all data.

predictor xm l/3<r<3 l/2<r< 2 2/3<X3/2 3/4<r<4/3 B/K-formula 0.91 100% 67% 56% 11% H/G-formula 0.98 100% 100% 89% 56% Stephenson 1.69 100% 44% 11% 11% Eq.(2.4) 1.55 67% 67% 44% 44% Eq.(2.5) 0.74 100% 100% 22% 22%

Table 4.3 Prediction potential for data in Table 4.1

predictor l/3<r<3 l/2<r< 2 2/3<r<3/2 3/4<r<4/3 B/K-formula 1.05 90% 70% 50% 10% H/G-formula 0.89 80% 70% 50% 30% Stephenson 1.88 90% 40% 30% 15% Eq.(2.4) 1.65 70% 65% 40% 30% Eq.(2.5) 0.78 95% 85% 35% 30%

Table 4.4 Prediction potenlial for all data in Tables 4.1 and 4.2

Evaluation of recent desk study results on granular filters Q24I6 May, 1998

Considering the results in Table 4.3 the prediction potential of the H/G-formula seems to be better. However, the differences are small and become less i f the less accurate values of the coefficients in the B/K-formula (see the remark in the beginning of this section) are taken into account (coefficients have been used as determined by Den Adel (1995). I f the data set is used as presented in Table 4.1 the prediction potential expressed in xm is close to 1: xm = 0.99.

The results in Table 4.4 for all data give the same result.

The result for the H/G-formula is much better compared to the earlier Hoffmans-formulas. For equation (2.8) a value of x„ = 0.33 could be determined (De Groot et al, 1997). Clearly, the differences are significant improvements.

The prediction potential of the other formulas is less. The Stephenson formula and equation (2.4) are unsafe predictors (a value f o r xm smaller than 1 means a prediction on the safe side

compared to the measured data).

Indeed, equation (2.5) predicts safe values (safer than the H/G-formula and the B/K-formula), but the accuracy for values of r between 2/3-3/2 and 3/4-4/3 is less.

Summarizing, the differences between the H/G-formula and the B/K-formula are very small. This may be expected, since both formulas were fitted on 9 experiments out of the same dataset. Taking into account furthermore the inaccuracies in the coefficients in the B/K-formula due to fitting on a slightly different data set, it may be concluded from the results that both formulas may be applied with a safety factor of 3.

The H/G-formula has the potential to be applied outside the experimental range for which it is validated. Physical properties such as degree of turbulence and particle gradation are defined in a statistically sound way. Also non-uniform flow may be introduced (see Section 3.6).

Finally, it should be noted that the B/K-formula as well as the H/G-formula are not validated for high turbulence levels and, therefore, should be applied within the limits of the experimental data.

Experiments with higher turbulence levels are recommended as no specific test data being available. Recently, the possibility to carry out this type of experiments and the suitability of various measuring devices have been shown (van Os, 1998; Spaargaren et al, 1997; Verheij,

1997).

WL I delft hydraulics

Evaluation of recent desk itudy results on granular filters Q 2 4 I 6 May, 1998

5 Conclusions and recommendations

This evaluation study enables the following conclusions:

• The prediction potential of the Hoffmans/Grass-formula and the Bakker/Konter-formula does not differ significantly, taking into account the limited number of experiments and the inaccuracies of the coefficients used for the B/K-formula due to fitting on a slightly different data set.

• Both formulas may be applied within the limits for which they have been validated, viz. uniform flow without high turbulence.

• In principle, both formulas may be extended to conditions with high turbulence and local flow conditions; however, they are not validated for it thus far. In particular, the H/G-formula has the potential due to its theoretical base, to be applied outside the experimental range for which it is validated. Physical properties related to degree of turbulence, particle gradation, different erosion levels, and non-uniform flow may be implemented relatively easily.

• The Stephenson-formula and the constant value formulas eq.(2.4) and eq.(2.5) have less predictive potential relative to the H/G-formula and the B/K-formula.

Definite answers about the applicability of both formulas for conditions with high turbulence levels can be obtained by validation on specific experiments. No specific test data being available, it is recommended to that these experiments be carried out. Emphasis in the experiments should be on the H/G-formula as this equation has a better theoretical basis than the B/K-formula.

Evaluation of recent desk study results on granular filters Q24I6 May, 1998

6

Literature

Adel, H. den, et al (1995): Implementatie filteronderzoek, Grondmechanica Delft, rapport CO349410/16, Delft

Akkerman, G.J. (1998a): Projectfdters, toplaagstabiliteit granulaire fdters benedenstrooms van constructies, WL|Delft Hydraulics, rapport Q2332, Delft.

Akkerman, G.J. and H.J.Verheij (1998b): Literatuurinventarisatie stabiliteitsvoorspellers en -gegevens. WL|Delft Hydraulics, verslag bureaustudie Q2395.10, Delft.

Bakker, K.J. et al (1994): Design relationship for fdters in bed protection, Journal of Hydraulic Engendering, A S C E Vol.120, no 9.

Booij, R. (1998): Erosie onder een geometrisch open filter, Technische Universiteit Delft, rapport 2-98, Delft.

Boutovskaia, L . (1997): Analyse gamma van experimenten M648/M863. Rijkswaterstaat, Dienst Weg-en Waterbouwkunde, internal note, Delft.

Breusers, H.N.C. en W.H.P. Schukking (1971): Begin van beweging van bodemmateriaal. Waterloopkundig Laboratorium, speurwerkverslag SI59-1, Delft.

Breusers, H N . C , 1972: Drukfluctuaties in turbulente stromingen. Waterloopkundig Laboratorium, verslag literatuuronderzoek S230, Delft.

CUR (1993): Filters in de waterbouw, CUR rapport 161, Gouda.

Compte-Bellot, G. (1963): Coefficients de dessymmétrie et d'apatissement, spectre et correlations en

turbulence de conduite, Journal de Mechanique, Vol.JJ, no.2

Grass, A.J. (1970): Initial instability of fine bed sands. J. of Hydr.Div., A S C E , HY3.

Groot, M.B. de, H.den Adel (1997): Keuze fdterregel voor bodemverdediging. Grondmechanica Delft, rapport CO-370780/18,Delft

Hauer, M. en T.van der Meulen (1996): Parallel stroming in een geometrisch open filterconstructie, een

theoretisch model voor de berekening van poriesnelheden en schuif spanningen, Technische

Universiteit Delft

Hauer, M. en T. van der Meulen (1997): Probabilistische analyse van onzekerheden bij ontwerp en uitvoering

van een enkellaags geometrisch open filterconstructie in een open waterloop, Technische Universiteit

Hoffmans, G.J.C.M. (1992): Two-dimensional mathematical modelling of local -scour holes. Delft University of Technology, Ph.D.Thesis, Delft.

Hoffmans, G.J.C.M. and R. Booij (1993): The influence of upstream turbulence on local-scour holes.Proc. 25th IAHR-congress, Tokyo.

Delft.

Evaluation of recent desk study results on granular f i l t e n Q 2 4 I 6 _{May, 1998}

Hoffmans, G.J.C.M. (1996a): Geometrisch open filters onder invloed van stroming en turbulentie. Rijkswaterstaat, Dienst Weg- en Waterbouwkunde, bureaustudie, Delft.

HofLmans, G.J.C.M. (1996b): Analyse Bakker/Konterformule. Rijkswaterstaat, Dienst Weg- en Waterbouwkunde, notitie, Delft.

HofLmans, G.J.C.M. (1997): Filterformule op basis van het schuifspanningsconcept. Rijkswaterstaat, Dienst Weg- en Waterbouwkunde, notitie, Delft.

Huijstee, J.J.A. van, en H.J. Verheij (1991): Verruiming ontwerpregels voor filters in bodemverdedigingen,

gelijktijdige instabiliteit van toplaag en filterlaag. Waterloopkundig Laboratorium , verslag

modelonderzoek Q572, Delft.

Nezu, I. (1977): Turbulent structure in open-channel flows Kyoto University, Ph.D.Thesis, Japan.

Nezu, I. and H. Nakagawa (1993): Turbulence in open-channel flows. IAHR-monograph, A.A.Balkema, Rotterdam.

Os, P. van (1998): Hydraulische belasting op een geometrisch open filterconstructie, Technische Universiteit afstudeerrapport, Delft.

Rodi, W. (1980): Turbulence models and their application in hydraulics, IAHR-Monograph, Balkema, Rotterdam.

Ruiter, J.C.C. de (1982): The mechanism of sediment transport on bed forms. Euromech 156: Mechanics of sediment transport, Istanbul.

Schukking, W.P.H. et al (1972): Systematisch onderzoek naar twee- en driedimensionale ontgrondingen, Waterloopkundig Laboratorium, verslag modelonderzoek M648/863, Delft.

Spaargaren, D. en H.J. Verheij, 1997: Haalbaarheidsstudie naar het meten van de demping van drukfluctuaties

in filterconstructies. Waterloopkundig Laboratorium, bureaustudie Q2231, Delft.

Shimizu, Y . , T.Tsujimoto en H. Nakagawa (1990): Experiment and macroscopic modelling of flow in highly

permeable porous medium under free surface flow. Journal of Hydroscience and Hydraulic

Engineering 1990 (8) No. 1 69-78.

Stephenson, D. (1979): Rockfdl in hydraulic engineering. Elsevier Scientific Publishing Company.

Verheij, H.J. (1993): Verruiming toepassingsmogelijkheden Bakker/Konter-filterformule. Waterloopkundig Laboratorium, notitie, Delft.

Verheij, H.J. (1997): Stroming in granulaire filters, oriënterende experimenten. Waterloopkundig Laboratorium, meetverslag Q2331, Delft.

WL | delft hydraulics

Evaluation of recent dole study results on granular filter» _Q24I6

May, 1998

A D e r i v a t i o n H o f f m a n s / G r a s s - f o r m u l a

Hoffmans derived a prediction formula for simultaneous instability of base and toplay material starting with the shear-stress concept as presented by Grass (Figure A - l ) . er

. Figure A.1 Shear stress concept according to Grass

Grass' concept of transport is based on statistical assumptions. The general idea is that transport will occur i f the strength of the soil is less than the load on the soil, as generated by the flow of water in an open channel. The load on the soil can be expressed as the shear force on the soil, whereas the strength of the soil can be expressed as the maximum allowable shear force on the soil, without transport of soil. The shear stress x will be a quantity which is suitable to define transport: transport will occur i f the maximum allowable shear stress on the soil is less than the actual shear stress originated by the flow of water. Since neither load nor strength are deterministic values, they are characterized by the distribution functions pff o )

(flow) and;?, (zs) (soil). Transport of material is proportional to the probability that transport

will occur:

Tec ƒ d rfPf( Tf) . jdT,p,(rs) ( A . 1 )

r , = 0 _r,=0

For the sake of simplicity each distribution function is characterized by an average value and a standard deviation. Instead of calculating these integrals, one can apply the concept of characteristic values.

vVU I delft hydraulics

Evaluation of recent desk study results on granular filters Q 2 4 I 6 _{May, 1998}

The characteristic value is a value which is higher or lower than the average value. Usually characteristic values are expressed in the average value and a fraction or manifold of the standard deviation. The characteristic strength of the soil is written as:

Tck = T

G

-YPc (A.2)

whereas the characteristic load is written as:

*o,k = t0+r,o-0 (A3)

A specific transport will occur i f r0.k - rf i* :

To+rlo-0 = T a - yl ac (A.4)

Assuming <Jc = acrG = acyc C ,A,pgDl50 the general formula follows:

•V>«o = T° ; (A.5)

Hoffmans (1992) presents relations for a0 for uniform and non-uniform conditions.

The amount of material transported depends on the value of y,. When y, is low, transport is high, whereas i f yx is high, transport is low.

Grass relates the strength of the soil to a y/G which is somewhat higher than y/s, the value as

derived by Shields:

Ts = Vc.sAtPSD.so (A.6)

Comparing equations (A.5) and (A.4) the relationship between Grass and Shields results:

1 + y,aa

Vc.G., = Vc,S,, (A.7)

With a0 = a0r0 = a0pu} substituted into (A.5) the stability equation for uniform flow will

be obtained: _ pul(l + aori) u'i^ " « r VcGjPgV-cc^y,)

Ll,

1 '

V (A.8) Wl. I delft hydraulics A - 2

Evaluation of recent desk stud/ results on granular filters Q24I6 May, 1998

Note that instead of y/c,s,i now y is the design variable representing the rate of erosion.

The above idea - comparing characteristic values for load and strength - has been used by Hoffmans (1997) for the transition layer between filter and base material:

T — T

* O.A:.transition ctk,transition

(A.9)

Analogous to Japanese research (Shimizu et al, 1990) which resulted in the following relationship for the velocity profile in a vertical direction within a granular layer:

"(*) = "bas, + (W, , o - M f c a » ) e XP ( * 1 * ) W i t h = A M 1 01 (A.10)

Hoffmans assumes a similar relationship for the shear stress in the toplayer (Figure A.2):

< z ) = *base + {r0 ~ tb a s i) exp(z / X) with A = Dt501 a (A. 11)

flow velocity top layer base material load T0 + Y°0 characteristic load Lo.k. transition transition

Figure A.2 Damping of shear stress according to Hoffmans (1997)

Applying Grass' concept, so replacing r0 by r0 + y, a0 leads to:

' 0,k,transition -a- _D,

ISO J

(A. 12)

' c.kjransition _{*c.c-Yb°-c,b = Tc.e-r}b<xc,bTc.G = rc, o ( l - 7 V *cJ = Vc^PgDhso^-Yb^cb)

(A. 13)

Substituting (A. 12) and (A. 13) with Tbase = TJ T0 into (A.9) results in:

Aso.

^0+ ( r0+ y , c r0- 7 7 z -0) e x p - a - ~ U yc^bpgAbDbS0(\-ybacb) (A.14)

Evaluation of recent desk study results on granular niters Q 2 4 I 6 May, 1998

and after re-arranging:

a

650 -a d

D _ISO

(A.15)

Multiplying both sides with Dl50'.

D, ISO _

= D<soVc.G,bPgK(\-Yb<xc,h

)-bSO -ad

L Aso J

(A. 16)

Substituting (A.5) for D,s0 at the rigth hand side of (A. 16) and some re-arranging finally

results in a general relationship:

D. ISO _ To+TiVo D, bSO ^0+(r0+y,cj0-r/T0)exp -ad Aso. \ - Y P c i ¥C.G.AI (A. 17)

This equation is a general formula with the implicit assumptions of Gaussian distributions of load and strength, and a relationship with the depth-averaged flow velocity.

Furthermore, substituting a0 = OQ z0 results in a formula restricted to uniform conditions:

D„ _{\ + ctoY, l-Y}bac,b ¥c,G.Ab

*50

77+ ( l + 77) e x p -a d

"A57.

l

-r,a

eJ YC.G.AI

(A. 18)

Assuming a.ib = a^, and yb = /, results in:

D, ISO _ 1 + Vc,G,b\ D, _bSO 7j + (l + a0y,-ri)exp -ad D, ISO

VCGA

(A.19)

Note that this expression does not depend on and so this relation is independent of the gradation of either toplayer or base material.

Formula (A. 19) can easily be changed by multiplying both sides with Dll5IDl50 in:

Evaluation of recent desk stud/ results on granular filters Q24I6 May, 1998 A . 5

_

A . 5 Yc.G.Ab -ad D, no (A.20)

For a one layer system (toplayer directly on base material) the subscript t may be replaced by ƒ resulting finally in the H/G-formula:

D / 1 5 D / 1 5 l + a0y, -ad A (50 ¥c.G,At (A.21) Wl I delft hydraulics _{A - 5}