@article{oai:mie-u.repo.nii.ac.jp:00006340,
 author = {林, 拙郎 and Hayashi, Setsuo},
 journal = {三重大學農學部學術報告 = The bulletin of the Faculty of Agriculture, Mie University},
 month = {Mar},
 note = {application/pdf, In this study the author dealt with the phenomenon of scour at the base caused by free falling nappe over sediment control dams, which is a vertical drop　structure. In attacking the problem of scour, the author examined the　deceleration of the flow which plunged into the scour hole, and the velocity of the flow along the bottom boundary. Based on the analyses, the author made a theoretical equation regarding the maximum scour depth, and an experimental equation regarding the maximum scour length. In the following, it was assumed that the bed material was uniform sand, and consideration was given to the penomenon of the scour which had reached equiliblium.
1. The parameter representing the scour phenomenon. 
Fushitani had been of the opinion that the scour reached equilibrium when the sediment discharge or transport rate out from the scour zone equaled zero. Noting a particle on the scour bed, however, the author considered the condition equilibrium for the scour as having the following explanations. When the scour reaches equilibrium, as the velocity at the bottom of plunging jet is reduced, it is thought that a grain of sand at the scour bottom remains motionless, and the scour is not enlarged.
The author assumed that a sand grain on the bottom remained motionless on reaching the condition equilibrium.
Then, the impinging velocity (ν₂) on the sand grain is defined as:
  ν₂/ν₀₀=ｋ(D/η)p  
　　　η=h/sin θ, h=T+hd   (1)
Where  ν₀₀ is falling velocity at the water surface on the scour hole, D is thickness of the nappe, θ is influx angle of the nappe, T is maximum depth of scour, h∆ is the average water depth in the vicinity of the falling nappe above the spillway of the counter dam, and k, p are constant. As the drag force and the friction force acting on the sand grain are in a balanced relation to one another, the equation for the maximum scour depth can be obtained by equating these two forces
 h/dsiθ= c[1/(so-1)tanψ(qν₀₀/gd²)(D/d)a]β          (2)       
　　　α=2p-1, β=1/2p                                                      (2’)
Where c is constant, s₀ is specific gravity of the grain, q is the discharge per unit of width, tan ψ is coefficient of static friction of the sand grain in the water, d is representative grain size, and g is acceleration of gravity. The function form of the equation (2) is the basic one to obtain the scour depth, etc., and so author gave the name Scour Parameter.
By examining the result of scour experiment for the case of free over fall, it was confirmed that this parameter became a very useful one for expressing the scour phenomenon. However, since the values of the exponent (α) and (β) obtained by this experiment did not satisfy the condition of the theoretical equation (2’) for α and β, it became necessary to consider the scour phenomenon again.
2. Hydraulics of the slanting submerged jet and the flow along the scour bottom
By the falling nappe flowing in water surface as jet, the water the nappe or jet will undergo both lateral diffusion and deceleration. This submerged jet flow will cause a two-dimensional diffusion. However, at the bottom of the scour hole this submerged jet flow will develop the flow of the other properties. This submerged jet, which has an impinged slant on the bottom boundary of the scour, comes to the stagnation point and then flows forward as a two-dimensional wall-jet. In the wall jet along the bottom boundary-layer is caused by viscosity of the water and by the mean velocity of the main flow, which is defined as the velocity of the mean flow outside the boundary-layer.
The mean velocity (u₀) of the main flow along the bottom boundary in the neighbourhood of the stagnation point can be found analytically by using the potential theory in this zone, and the simple equation of this solution is finally written as: u₀/u₀ max = Aξ/ξ2 + Bξ+C’ ξ=x/h sinθ　           (3)
Where A,B,C are constant, and x is the length along the bottom boundary from the bottom boundary from the stagnation point. 
The mean velocity  (u₀) of the main flow along the bottom boundary was　measured the value of coefficients of the equation (3). In this experiment, falling nappe was two-dimensional, and one of the bottom boundaries was the rough flat plate, another was the rough boundary. The results showed that u₀ max was expressed by Albertson et al. , equation for the two-dimensional submerged jet. Depending on whether the boundary of the bottom was the plate or the fixed scour boundary, values of the coefficients in the equation (3) differed. For the case of the fixed scour boundary, the following values of the coefficients A, B, C in the equation (3) were obtained
     A=0.600, B=-0.284, C=0.194                   (4)
The velocity (u₀) of the main flow causes the velocity distribution in the　boundary-layer along the bottom by traction along the lower layer. The equation of the thickness (δ) for the boundary-layer was derived by using Krmn’s momentum-integral equation for two-dimensional boundary-layer, and in this equation,  (height) to a low power(n) was used to determine velocity (u) in the boundary layer. The values of coefficients n,  λin this equation for the thickness were unknown. However, by comparing the theoretical values with experimental values, the author obtained:
       n=1/6, λ=0.0125
These values agree with the usual n, λfor turbulent flow a rough plate.
On the one hand, shearing stress (τb) on the bottom was determined finally by obtaining n and λ so that the theoretical value thickness (δ) for the boundary-layer agreed with the experimental value. It was shown that this τb curve had its peak at a certain point (ξ), whereξ was the dimensionless length along the boundary, and the value of this point was approximated asξ=0.4. When a set of coefficients A, B, C for a flat boundary was used in the equation (3), the equation for the thickness (δ) and shearing stress (τb) should hold proportionately for the condition of the fixed scour boundary.
          3. Maximum scour depth reaching equilibrium
The author obtained the following equation for u₀ max through measuring the velocity u₀ of the bottom flow for the actual scouring on reaching equilibrium:    　　 　　　uo max/ν00=k(D/η)p, p=3/4                                                           (5)
By using u₀ max obtained from the above equation, the author investigated the coefficients A, B, C of the equation (3) for the scour equilibrium, and it became clear that the values of the equation (4) for fixed scour boundary were applicable as the values of the coefficients A, B, C in the scour experiment.  Thus, it permitted τb and τb max for a scour condition which had reached equilibrium to be expressed by the equation for the fixed boundary. But it was considered that exponent (n) of the velocity distribution for the boundary-layer in this case differed from the exponent in the fixed boundary.
By using the τb max stated above, the author obtained the following equation from the balance of the grain on the scour bottom which had reached equilibrium   h/d sin θ=c[1/(so-1)tanψ(qυ00/gd2)(D/d)α]β
α=2p-1, β=1/(2p +γ), h=T + hd, γ=2n/(1+2n) 　　 (6)
Where n is the exponent for the velocity distribution in the boundary-layer.　Although equation (6) is observed to be the same from as equation (2) as derived in section 2, the exponent (β) in equation (6) contains the factor (n) regarding the boundary-layer, and in this respect equation (6) differs from equation (2).
If possible the scouring test is performed with D/d varied artificially so that the coefficients α, β, c of equation (6) are determined. Thus, the special device of jet flow has been made for this scour experiment, and from the experimental results of scour with this device, the following value of αand βhave been obtained
           α=1/2, β=1/2                        (7)
By substituting equation (7) for α=2p-1 in equation (6), the value of p becomes:  
             p=3/4                    (8)
On the other hand, as shown in the equation (5), the same value for p has already been obtained, and as a result, the relation betweenα and p agrees with the author’s theory.
The theoretical relations for the other exponent (β) in equation (6) are:
             β=1/(2p+γ), γ=2n/(1+2n)            (9)
The exponent (n) of the velocity distribution for the boundary-layer, therefore, substituting equation (7) and (8) in equation (9) yields:   
             n=1/2        (10)
This value of n will become greater than the value (n=1/6) for a fixed boundary, because the velocity gradient in the neighbourhood of the bottom boundary in the case of actual scouring is smaller than the value of that for a fixed boundary.
Furthermore, since Yamaguchi found that the exponent (n) of the velocity distribution is n=1/2 in case of turbulent flow for the hydraulic rough regime in which the slope and steep and the ratio of the water depth to the height of the roughness element is small, the value of equation (10) in this case is rather appropriate.
From the above explanation, therefore, the values of p and n for experimental results nearly agreed with theoretical results. As result, it became clear that the formula relating to the exponent (α), (β) in equation (6) for the scour depth was valid. By substituting the experimental value for scour depth in equation (6), the author determined the value of the multiplication constant (C) which remained to the last.
The problem with the exponent (α),(β) as stated in section 1 could be solved by the theoretical equation n the basis of the scour parameter using the boundary-layer theory, and hence it was shown that this theory explained the scour phenomenon in a universal way. In this study, furthermore, the furthermore, the formula for the scour length was obtained by the application of the scour parameter.},
 pages = {101--189},
 title = {砂防ダム水叩部の洗掘現象に関する水理学的研究},
 volume = {66},
 year = {1983}
}