CALCULATION METHODS ON CRITICAL DETACHMENT OF NON-UNIFORM GRAVEL FROM STREAM BED

Fangyi Wang

Laboratory of Hydraulic Engineering

3-17-2 Kamitsuruma, Sagamihara-shi, 228-0802 Japan

Tel. +81-42-748-2921, Fax. +81-42-748-4714 

Abstract: Methods of prediction of critical bed shear stress for the detachment of non-uniform gravel from stream bed were examined by the application of the relationship derived for the spherical particle.  In the process of calculation as a way of determination of the term related to resisting forces included in the relationship for the sphere, an equivalent sphere to the actual gravel is considered.  An equivalent equation is considered and used in combination with the relationship for a sphere (d_i) contacting in ideal condition (contacts with two bed-spheres (d_b) in downstream side and protrude from them in arbitrary protrusion rate).  The coefficients included in the equivalent equation are determined so that the calculated results by the relationship for the sphere coincide with the criterion relationship which fit in the existing data. In the cases of calculation, two types of equivalent equation and two kinds of region dividing are adopted for comparison.

Before discussion of the subject in this paper, summary of the process of calculation and important parts of associated relationships are shortly explained. 

Keywords: critical bed shear stress, critical detachment, non-uniform gravel.

1    INTRODUCTION

In reference [3], a relationship on the critical detachment for spherical particles was obtained. In the check of its validity, the experimental results on the spheres of various diameters resting on three bed spheres were used. In [4], the equivalent equation was considered in order to apply the relationship for the sphere to actual gravel, and the calculated results of the dimensionless critical bed shear stress based on the equivalent equation were compared with Egiazaroff's formula. In [6], two types of equivalent equation and two kinds of division method of region were treated, and Hayashi et al's formulas were adopted instead of Egiazaroff's formula as a criterion of comparison. In both [4] and [6], the relationship on the critical detachment for the sphere divided by a shelter coefficient was used for comparison with the criterion relationship. However, the shelter effect is not considered in this paper. In such a case, the picture of the acting forces becomes clearer than the previous case.  Because the calculation process used in the past paper is basically adopted in this paper, important parts of associated relationships are given before the sections on the subject.

2    RELATIONSHIP OF CRITICAL DETACHMENT FOR SPHERICAL PARTICLES

 In the case of an isolated sphere of arbitrary diameter d_i which contacts with two bed spheres of diameter d_b in down stream side and protrudes from them with arbitrary protrusion rate h_i =S_ei/d_i (Fig.1), from the equation of moment equilibrium due to fluid forces and the submerged weight of the sphere of di, the critical bed shear stress for detachment can be expressed in dimensionless form as follows [3] 
 (The position of Fig.1)

t _*cdfi = t _*coi /K_dfi                                     (1)

t _*coi =1/[(4/3)(u_o1i /u_*ci )²C_D1i(G_i /ma_i )]                  (1-1)

K_dfi = (e_DAi /e_Di)(G'_Ai /G_i )(1+n_D1i ?C_vD1i )               (1-2)

where, t _*cdfi = u_*ci ²/gs'd_i, s'= 1-s, s = r_s / r, u²_*ci =gh_ci sina, h_ci = h_oci +ad_b,

a = 0.3 [2] ; u_o1i /u_*ci = 5.75 log[ 30.1 (a + 0.5r_D) ], r_D =d_i /d_b ;

C_D1i =F_D1i / [(1/2) ru_o1i ²( p/ 4)d_i ² ] ; G_i / m a_i =k_Li +k_Di /m a_i, k_Di =F_Di / F_D1i,

k_Li =F_Li / F_D1i ; m a_i =m _i cosa-sina, m _i = tanf _i = e_Li /e_Di  (e_2i /e_1i was noted in [3])

Notations : t _*cdfi : dimensionless critical bed shear stress for a sphere of d_i (suffix d and f refers respectively to the effect of deviation from the sphere center and the fluctuation of the fluid forces) ; h_ci : the flow depth measured from the origin (a distance ad_b under the top of bed spheres) at critical condition, h_oci : distance from the top of the bed spheres to the water surface ; d_b : the diameter of bed sphere ; a : a factor to d_b , a = 0.3 ([2], [3]) ; a. : the inclination angle of stream bed from horizontal line ; g : the gravitational acceleration ; r_s and r : the density of the sphere and water ; u_o1i : the time averaged velocity at the sphere center of diameter d_i fully protruding from the bed spheres (h_i =S_ei /d_i =1) ; S_ei : distance from the top of the sphere (d_i) to the top of the bed spheres (d_b) ; r_D : the relative diameter ; k_Di and k_Li : the relative drag and lift forces ; F_Di and F_Li : the time averaged drag and lift forces at arbitrary protrusion rate ; C_D1i : the drag coefficient to F_D1i ; F_D1i : the time averaged drag force at a sphere of d_i fully protruding from the bed spheres (at h_i =1) ; f _i : the pivoting angle of the sphere d_i.

 K_dfi may be interpreted as a coefficient reflecting the combined effect of the deviation and fluctuation of the fluid forces (refer to [3] about the notations included in K_dfi ).  Further, for a sphere of d_i situated as in Fig.1, the following relationships on f _i = tan^–1m _i, h_i and r_D can be obtained by the geometrical calculations [3].

h_i = [ ( r_D² + 2r_D )^1/2sinb _i + r_D-1] / 2r_D                 (2-1)

f _i =90°-b _i ; sinb _i = ( 2r_D  h_i + 1-r_D ) / ( r_D² + 2r_D )^1/2          (2-2)

3    CRITERION RELATIONSHIPS

In Fig.4 of [5], a lot of data on the critical bed shear stress for the detachments are plotted.  Relationships of Egiazaroff t _*ci / t _*cb = [ log19 / log (19 r_D) ]² (for total region) and those of Hayashi et al, t _*ci / t _*cb = r_D ^–1.0  (r_D 1.0), and t _*ci / t _*cb = [ log8 / log (8 r_D) ]²   

(r_D >1.0) are also indicated, where suffix b refers to arithmetic mean size of the grains.  At a glance of the figure, it is seen that the relationships of Hayashi et al fit in the plotted points in wide range.  Therefore, we adopted the following relationships as the criterion to check the validity of our calculation methods.

That is :   t _*ci / t _*cb = r_D ^–1.0  (r_D 1.0) (c-1) and

         t _*ci / t _*cb = r_D ^–0.7  (r_D >1.0) (c-2).

Eq.(c-1) is that of Hayashi et al in itself and Eq.(c-2) is the approximation of the relationship of Hayashi et al in region r_D >1.0.

4    PROCESS OF CALCULATION

To apply the Eq.(a) for the spheres to the actual gravel, for convenience, the spheres situated at ideal resting condition as in Fig.1, and having equivalent resting effect to the gravel are considered. The equation of the equivalent effect (tentatively is called as equivalent equation) is used together with Eq.(b). The coefficients included in the equivalent equation are determined so that the calculated results by t _*ci = t _{*cd f i} coincide with the criterion relationships.

5    RELATIONSHIPS FOR TERMS ON FLUID FORCES

The terms of C_D1i, k_Di, k_Li and K_{d f i} included in Eq.(a) are supposed to be depend on h_i , r_D and so on.  The function forms and coefficients used in [6] are adopted here also as follows

5.1    For the region h_i >0

C_D1i = 0.7 k_H , where  k_H = 1 - 0.74h_i   (0<h_i 1.0) (1)

k_Di = a_Di  h_i + b_Di  where  a_Di = 1 - b_Di  (0 <h_i 1.0),

b_Di = -0.14r_D + 0.21  (r_D 1.0) , 

b_Di = 0.0869 / exp(0.2163r_D)  (r_D >1.0)(2) ; 

k_Li = a_Li  h_i + b_Li  where a_Li = (0.3 - b_Li ) / 0.8  (0<h_i 0.8),

b_Li = -0.1r_D + 0.20  (r_D 1.0),  b_Li = 0.1292 / exp(0.2559 r_D) (r_D >1.0), 

k_Li = -2hi + 1.9 (0.8<h_i 1.0)(3)

K_{d f i} = 2.4  (0<h_i 1.0) (4)

5.2    For the region?h_i 0

C_D1i = 0.7 k_H , where  k_H = 1.0(5) ;

k_Di = 1 / [1 + m_Di  exp (-m_Di'  h_i )] , where  m_Di = (1 / b_Di ) -1.0,

b_Di = -0.14 r_D + 0.21  (r_D 1.0) ,  b_Di = 0.0869/exp(0.2163 r_D) (r_D>1.0),

m_Di' = 3 r_D(6) ;

k_Li = 1 / [1 + m_Li  exp (- m_Li'  h_i )] , where m_Li = (1 / b_Li )-1.0,

b_Li = -0.1 r_D + 0.20  (r_D1.0) , b_Li = 0.1292 / exp(0.2559 r_D) (r_D>1.0),

m_Li' = 3 r_D(7) ; K_{d f i} = 1 + 1.4 / exp (-m_{d f}'  h_i ) ,where

m_{d f}' = 3 r_D(8)

notes :

(1) The function forms of Eqs.(1), (2) and (3) for the region h_i >0 are assumed from  some simulations as well as some experimental results on uniform spheres ([2], [3], [6]). 

(2) The function forms of Eqs.(6), (7) and (8) for the region h_i 0 are assumed from some inferences and simulations ([3], [6]).

(3) The relationships Eqs.(14b), (15b) and (16b) on kDi, kLi and K_{d f i} for the region h_i 0 on page 130 in [3] corresponding to our Eqs.(6), (7) and (8) mentioned as above are partly misprinted.

6    COMPARISON OF CALCULATED RESULTS  WITH CRITERION RELATIONSHIPS

Two kinds of equivalent equations : f _i = a' r_D ^b' (d) [1],[4] and h_i =kar_D ^km+k_b(e) [6] are used together with Eq.(b) respectively to obtain the m _i values in Eq.(a). In selection of the regions, the variable type is also tried in addition to the fixed type. In the calculation hereafter, G_i / m a_iG_i / m _i = k_Li +k_Di / m _i is used as an approximation.

6.1    Application of eqs.(D) and (B) to fixed regions

 We adopt r_D = 0.1~1.0 and r_D = 1.0~10 as two fixed regions, and the case of t _*ci = t _*cb Eq.(c) for t _*cb = 0.06 is considered as the first example.

 For a selected region, the coefficients a' and the exponent b' in Eq.(d) can be obtained so that the results by t _*ci = t _{*d f i}(a) coincide with those by t _*ci = t _*cbEq.(c) for t _*cb = 0.06 at two ends of the selected regions. Thus the following relationships are obtained.

f  _i = 72.88 r_D ^–0.06681  (0.1, 1.0)                    (9-1)

f  _i = 72.88 r_D ^–0.2285   (1.0, 10)                     (9-2)

The numbers in parenthesis are rD values at two ends of the regions. The results by Eq.(9) are shown in Fig.2 by the broken lines and those by t _*ci = t _*cbEq.(c) for _*cb = 0.06 are by two solid lines with different inclinations. It is seen that inside the two ends of the regions the broken lines deviate considerably from the solid lines. (The position of Fig.2)

6.2    Application of eqs.(E) and (B) to fixed regions

For a selected region, the k_a, k_b, and k_m in Eq.(e) can be determined by trial calculations so that the results by t _*ci = t _{*cd f i} coincide with those by t _*ci = t _*cbEq.(c) for t _*cb = 0.06 at three points. Among those, two are at the ends of the region and another is at a point selected freely between the other two, so that the general view of the degree of agreement become best.  Thus for the case t _*ci = t _*cbEq.(c) for t _*cb = 0.06, the following relationships are obtained.

h_i = -0.3182 r_D ^–1.185 + 0.573  (0.1, 0.4, 1.0)             (10-1)

h_i = -1.045 r_D ^–0.3659 + 1.30    (0.1, 3.0, 1.0)            (10-2)

The results by Eqs.(10) are shown in Fig.2 by the dotted lines.  Even though the deviations from the solid lines are considerably small as compared with broken lines, but it is still remarkable especially at r_D 0.5~1.0. This trend may be attributed to the fact that the values of C_D1i, k_Di, k_Li and K_{d f i} change abruptly at h_i = 0 and h_i = 0.8. In view of such a fact, the variable regions are adopted in the next subsection.

6.3    Application of eqs.(E) and (B) to variable regions

We adopt the r_D values corresponding to f _i = 90°(m _i = 8), h_i = 0, h_i = 0.8 and t _*cb = 0.06 as the division points for the variable regions. These division points are determined by the trial calculation so that the calculated results by t _*ci = t _{*cd f i} coincide with the results by t _*ci = t _*cbEq.(c) for t _*cb = 0.06. For such regions determined this way, the following relationships are obtained.

For t _*cb = 0.06 : h_i = -0.2341 r_D ^–1.295 + 0.393  (0.081, 0.3, 0.67)       (11-1)

h_i = -0.995 r_D ^–0.5698 + 1.25  (0.67, 0.8, 1.0)          (11-2)

h_i = -0.815 r_D ^–0.5028 + 1.07  (1.0, 3.0, 9.0)           (11-3)

h_i = 0.05 r_D ^1.0 + 0.35  (9.0, 9.5, 10)                (11-4)

The numbers of 0.081, 0.67, 1.0 and 9.0 in parenthesis are the r_D values corresponding to f _i = 90°, h_i = 0, t _*cb = 0.06 and h_i = 0.8, but the number 10 is fixed.  Other numbers in the middle in parenthesis of Eqs.(11-1) ~(11-4) are the r_D values selected arbitrarily so that the general view of the degree of agreement become best. The calculated results by Eqs.(11) are shown in Fig.3 by the upper dotted lines and the results by t _*ci = t _*cbEq.(c) for t _*cb = 0.06 are by the upper solid lines. The degree of agreement is improved considerably than the case of fixed regions. According to the similar process, the relationships for t _*cb = 0.05 and 0.04 can be obtained as :  (The position of Fig.3)

For t _*cb = 0.05 : h_i = -0.2474 r_D ^–1.256 + 0.47 (0.065, 0.2, 0.6)        (12-1)

h_i = -4.926 r_D ^–0.1247 + 5.25 (0.6, 0.8, 1.0)          (12-2)

h_i = -0.771 r_D ^–0.5520 + 1.095 (1.0, 2.5, 5.7)         (12-3)

h_i = -29.03 r_D ^–3.065 + 0.94  (5.7, 8.0, 10)          (12-4)

The results in such a case are shown in Fig.3 by the middle dotted and solid lines.

For    t _*cb = 0.04 : h_i = -0.2409 r_D ^–1.245 + 0.536 (0.05, 0.15, 0.526)              (13-1)

h_i = -9.584 r_D ^–0.06614 + 10 (0.526, 0.7, 1.0)                   (13-2)

h_i = -0.704 r_D ^–0.6266 + 1.12 (1.0, 2.0, 3.52)                   (13-3)

h_i = -2.711 r_D ^–2.229 + 0.964 (3.52, 6.0, 10)                   (13-4)

 The results are also shown in Fig.3 (lower two lines)

7    CONCLUSIONS

(1) For a fixed region of r_D, the Eq.(e) as an equivalent equation is better than the Eq.(d) because the Eq.(e) has more one coefficient than Eq.(d).

(2) Application of Eq.(e) to the variable regions of r_D is capable of increase in the degree of agreement of the comparison with the criterion relationships, even though the calculation process becomes somewhat troublesome.

(3) In our calculation process as above mentioned, the forms of relationships on k_Di and k_Li, especially for the region h_i >0 and the coefficients b_Di and b_Li included in them are determined from somewhat rough assumptions and simulations. Their  adequacies need to be verified by the experiments or other ways.

References

[1]    Miller, R. L. and Byrne, R. J. : The angle of repose for a single grain on a fixed rough bed, Sedimentology, Vol. 6, 1966, pp. 303-314.

[2]    Wang, F. Y. : Critical detachment of protruded spherical particle from bed particles, Proc. of 6th International symposium on River Sedimentation, New Delhi, India, 7-11, November 1995,  pp. 659-669.

[3]    Wang, F. Y. : Critical detachment of a sphere of arbitrary diameter protruded from bed spheres at various protrusion rate, Proc. of 10th Congress of APD of IAHR, Langkawi Island, Malaysia, 26-29, August 1996, Vol. 2, pp.125-132.

[4]    Wang, F. Y. : Critical detachment of spherical particles of various diameters protruding from bed spheres and the application, Proc. 51st Annual Convention, JSCE, September 1996,  Part 2, II-277, pp. 554-555. (in Japanese)

[5]    Patel, P. L, and Ranga Raju, K. G. Critical tractive stress of non-uniform sediments Journal of hydraulic research, Vol. 37, No.1, 1999, pp.39-58.

[6]    Wang, F. Y. Calculation methods on critical detachment of non-uniform gravel from stream bed, Proc. of 55th Annual Convention, JSCE, September 2000, Part 2, II-228, pp. 456-457. (in Japanese)