The author\u2018s\u00a0own research project, dealing with morphological changes in small, relatively steep upland streams under high flows, was carried out in two stages:<\/p>\n
- \n
- Field investigation with monitoring the river channels\u2018 reaction to natural or artificially induced morphological changes. The aim of this stage was to\u00a0perform a\u00a0qualitative description of the changes.<\/li>\n
- Laboratory model experiments, which were focused on the design of\u00a0a\u00a0quantitative model of morphological transformations with the ability to predict the morphological response of the river channel when using environmentally friendly and cost-effective restoration measures.<\/li>\n<\/ul>\n
![\"\"](\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\")
<\/a>\n<\/a>\nFig. 1. Development of the \u201epear shaped\u201c scour resulting from degradation of a\u00a0drop structure on flash flood (left); feature of flow in transitional zone of the spatial 3D scour (right)<\/h6>\n
In a\u00a0part of the project with the aim of enabling the quantification of morphological changes, systematic research was carried out in laboratory conditions on two different types of channel model in the transition zone. For a\u00a0more complex description of the model behaviour, a\u00a0\u201ecomplete 3D\u201c model of the river channel in the transition zone of its opening was chosen (Fig. 2 left<\/em>). In order to speed up the process of documentation of the immediate development of the\u00a0scour hole depression of the river channel, a\u00a0model of a\u00a0symmetrically simplified \u201ehalf\u201c river channel was designed (Fig. 2 right<\/em>). The axis of channel symmetry was the\u00a0glass wall of the hydraulic channel, in which the channel was created in the entrance lined part and the transition open part. This second model made it possible to capture the course of the water level and river bed very quickly in the longitudinal direction almost immediately. To speed up geometric alignment of the water level and river bed, optical accessibility through the glass side wall of\u00a0the channel was used. It was also possible to measure the velocity fields in the\u00a0transition zone of the river channel significantly faster than with the complete 3D\u00a0model of the river channel. It is obvious that the assumption of symmetry was also a\u00a0simplifying assumption from the point of view of the real development of\u00a0the river channel in the opening, which had to be verified in the next phase of the project with a\u00a0complete 3D river channel model. In principle, there should be no objective reasons for significantly asymmetric scour hole development, unless special conditions are created. Further details of the research are described in the literature [5, 6] and are not presented here for brevity.<\/p>\n
<\/a>\nFig. 2. Full space model of the channel in the transition zone of scour hole development (left); symmetric half-space model of the channel with axis at the flume glass wall (right) \u2013 all under laboratory conditions<\/em><\/h6>\n
Scour hole development in a\u00a0complete channel model<\/h3>\n
Based on the observation of the successive development of the 3D scour hole in non-cohesive materials, partial findings can be summarized as this (see\u00a0diagram in Fig. 1 right<\/em>):<\/p>\n
- \n
- At the transition from channel section with a\u00a0fixed boundary to section with a\u00a0loose boundary, the whole process begins with the formation of a\u00a0small erosive hole in the bed, as also happens with wide channels. When a\u00a0sill is spontaneously formed in the bed at the transition between the channels, a\u00a0flow with the character of a\u00a0hydraulic jump arises. At first, it is a\u00a0flow with an\u00a0undular or weak hydraulic jump.<\/li>\n
- As the depth of the bed depression increases, this flow in the longitudinal level does not significantly change its character. As soon as the bed depression reaches the foot of the slopes, their stability is disturbed and small parts or even whole blocks of material start to slide into the depression. The\u00a0slopes of the changing channel near their foot no longer smoothly follow the slopes of the lined channel. Here, the current breaks away from the wall and the first lateral vortices are formed at the foot of both slopes.<\/li>\n
- Lateral vortices begin to gradually grow and become stronger. They get the\u00a0majority of their circulation energy from the main stream near the channel axis. The side holes expand from both sides, compressing the central stream and leading to its lateral contraction. This multiplies its tangential shear effect at the bed, and from this moment the deepening of the scour hole proceeds very quickly.<\/li>\n
- The more the scour hole extends to the sides, the more developed the\u00a0circulation structure of the side holes is. The sand material, carried from the\u00a0bed by strong current of the hydraulic jump near the longitudinal axis of the channel, is partly carried away from the scour hole area and partly circulated in the side holes. The movement of individual sand particles in the circulation areas takes place from the centre of the stream to the sides towards the slopes; from here along the side slopes it is sucked upstream and\u00a0back into the main stream in the channel axis.<\/li>\n
- Once this circulating sediment process stabilizes, further development of\u00a0the\u00a0maximum scour hole depth begins to slow down. The main stream already has a\u00a0partially exhausted transport capacity with the material that is circulated in the side holes. This material re-enters the main stream whenever the flow in the side holes moves it at the bottom to the channel axis.<\/li>\n<\/ul>\n
Sometimes the width of the side holes becomes too large for the back-current along the slopes to have sufficient force to remove the material from the\u00a0bed. The scour hole deepens more near the centre; at the edges it deepens less or not at all. In such cases, the bed at the edges of the scour hole depression is very flat, and at the centre of the channel it slopes steeply to the point of maximum depth of the scour hole. Some kind of side platforms are created here with one narrow deep bed hole in the middle of the profile.<\/p>\n
Hydraulic basics of scour hole study<\/h3>\n
Since a\u00a0hydraulic jump occurs in the place of a\u00a03D scour hole, the characteristics of which can be obtained by applying momentum theorem, it is quite natural to consider that the model of a\u00a03D scour hole will also be based on this principle. Momentum theorem allows an exact solution when designing the dimensions of constant-width apron in weir, but for the case of a\u00a0\u201epear\u201c 3D\u00a0scour hole the width is variable. It is also necessary to take into account the\u00a0normal pressure force, which is the reaction of the walls of the \u201epear\u201c object, although in the\u00a0case of deformable banks this does not apply either. In plan view, the\u00a0dimension of the hydraulic jump from the sides is not defined by solid walls, but by the interfaces with the lateral votices (Fig. 1 right<\/em>). Due to the fact that the water has a\u00a0peripheral velocity at the interface with lateral vortices with vertical axis, which is determined by the intensity of the flow circulation in the\u00a0vortex, it is necessary to calculate not only the magnitude of the normal force, but also the\u00a0tangential shear force. In addition, the distribution of pressures in the inlet profile may not always be hydrostatic, and the momentum of the flow in the\u00a0outlet profile is also affected by the size of the Boussinesq number (the\u00a0ratio of the actual momentum of the flow to the momentum expressed from the cross-sectional velocity) due to the non-uniform distribution of velocities in the cross section profile. It is clear from the above that a\u00a0simple application of momentum theorem does not achieve the objective. The author\u2018s\u00a0proposal appears to be more promising; following the example of Hunzinger [4], he makes the dimensionless length of the wake in wide channel section:<\/p>\n
<\/a>\nand the parameter F of the kinetic to potential energy exchange due to flow expansion (Eq. 2<\/em>).<\/p>\n
<\/a>\n(2)<\/p>\n
The dimensionless parameter F was defined by Ashida [1] by the following relationship:<\/p>\n
<\/a>\n<\/p>\n
<\/p>\n
which is an expression of the energy exchange in the transition zone between the input profile 1, where the original width of the channel is unchanged, and profile 2 where, in contrast, the width of the scour hole to the sides is the largest. The reason why the energy exchange parameter F could be a\u00a0good indicator of the tendency to create a\u00a0\u201epear\u201c 3D scour hole stems from the consideration that the more fluid\u2018s\u00a0energy (kinetic and potential) is converted in the\u00a0balance cross section into a\u00a0dissipative form of energy and into a\u00a0part of the energy applied during the transport of river channel bed and banks material, one can expect greater transformative effects of the current on the channel morphology, and thus a\u00a0greater spatial extent of the created scour hole. All quantities appearing in relationship (Eq. 3a<\/em>) are shown in Fig. 3<\/em>. In this diagram, the meaning of the given values is as follows:<\/p>\n
V1<\/sub>, V2<\/sub> [m.s-1<\/sup>] \u2013 cross-sectional velocities in the respective profile<\/p>\n
A1<\/sub>, A2<\/sub> [m2<\/sup>] \u2013 flow areas in the relevant balance profile<\/p>\n
c .A2<\/sub> [m2<\/sup>] \u2013 part of the flow area in profile 2, reduced by the recirculation area of the lateral vortices<\/p>\n
H1<\/sub> [m] \u2013 flow depth in balance input profile 1<\/p>\n
hp1<\/sub>, hp2<\/sub> [m] \u2013 water stage at the entrance to the scour hole and in\u00a0the scour hole respectively, measured from the\u00a0lowest level of the bottom of the scour hole<\/p>\n
\u2206HP<\/sub> = hp1<\/sub> – hp2<\/sub> [m] \u2013 difference in the water stages in the balance profiles<\/p>\n
Ys<\/sub> [m] \u2013 \u201epear\u201c scour hole depth<\/p>\n
Lw<\/sub> [m] \u2013 total \u201epear\u201c scour hole length<\/p>\n
B1<\/sub>, B2<\/sub>, B3<\/sub> [m] \u2013 stream widths at the surface at the entrance to\u00a0the\u00a0scour hole, in the scour hole, and just behind the scour hole<\/p>\n
In Eq. 3a<\/em>, however, a single cross-sectional velocity is considered in the entire profile 1 and 2. Based on the author own research on the distribution of point\u00a0velocities in the transition zone of scour hole development [5], it is recommended to modify the cross-sectional velocity elements in profiles 1 and 2 with the respective profile kinetic energy coefficients (Coriolis number \u03b1e), including the ratio of the actual kinetic energy height to the energy height expressed from the mean cross-sectional velocity.<\/p>\n
<\/a>\n<\/p>\n
In the balance relationship (Eq. 3b<\/em>), special care must be taken to choose the\u00a0appropriate position of both balance profiles \u2013 especially profile 2 (Fig. 3<\/em>).<\/p>\n
<\/a>\nFig. 3. Typical form of the \u201cpear\u201d shaped scour hole in a\u00a0long section (left) and a\u00a0plan view (right) with basic dimensions marked at characteristic profiles of scour hole<\/h6>\n
Methodology for determining channel changes<\/h3>\n
The main goal of this paper is to propose, based on previous observations [5, 6], a\u00a0simple procedure for determining the dimensions of a\u00a03D scour hole, which occurs at the transition from a\u00a0lined channel section with fixed boundary to a\u00a0channel with completely loose boundary without any technical support. The\u00a0procedure should be as simple as possible, making it possible to determine the extent of morphological changes in the river channel without the need to know too many details, which can mostly only be determined in laboratory conditions.<\/p>\n
Relationships for determining the basic dimensions of\u00a0a\u00a03D scour hole<\/h3>\n
The measurement procedure on the \u201efull\u201c 3D model was very slow and tedious. It was not possible to measure the corresponding time courses of the\u00a0water level and the bottom in the longitudinal profile and in both balance cross profiles. For systematic measurement, it was therefore necessary to limit access to a\u00a0half channel. All evaluated experimental data in the graphs in Fig. 4<\/em> come from these measurements.<\/p>\n
<\/a>\nFig. 4. Relationships derived to design a\u00a0preformed scour hole at a\u00a0channel transition; Graphs 1\u20136 are referred in the flow chart of design procedure of the scour hole (Fig. 6<\/em>)<\/h6>\n
It is not very practical for a\u00a0river engineer to use a\u00a0formula in the form of\u00a0Eq.\u00a03b<\/em>, for example due to a\u00a0lack of knowledge about the distribution of local velocities in profile 1 and especially 2 (Fig. 3<\/em>), where the lateral vortices arise. Some alternate, more straightforward relationships need to be used. The relationship between F<\/em> a\u00a0\u03b2 = B2<\/sub>\/B1<\/sub> plotted in Fig. 4<\/em> (Graph 1<\/em>) is presented. It is clear that F<\/em> is not only a\u00a0function of the geometric dimensions of the scour hole, but is also related to the properties of the flow before entering the changing channel. Therefore, the following relationship is proposed: F<\/em> = F<\/em>(\u03b2, Fr<\/em>1<\/sub>), where Fr<\/em>1<\/sub> is the Froude number of the flow in the inlet section 1 defined by the relationship Fr<\/em>1<\/sub>\u00a0= V1<\/sub>\/(gH1str<\/sub> )0.5, where H1str<\/sub> is the mean flow depth H1str<\/sub> = A1<\/sub>\/B1<\/sub>, A1<\/sub> is the flow area in the balance profile 1 and B1<\/sub> is the width of the surface flow in the same section. An exponential relationship (Eq. 4<\/em>) was derived, which naturally fulfils the logical condition F<\/em> = 1 for the channel widening ratio \u03b2 = B2<\/sub>\/B1<\/sub> = 1.<\/p>\n
<\/a>\n<\/p>\n
The relationship obtained by interpolating experimentally determined points reaches the coefficient of determination R2<\/sup> = 0.867.<\/p>\n
Furthermore, the relationship between the dimensionless flood length \u03bbw was obtained from the experimental data (Fig. 4<\/em> \u2013 Graph 2<\/em>) in analogy with Hunziger relationship (Eq. 2<\/em>). The relationship takes into account the\u00a0dependence of the\u00a0geometric dimensions, participating in the expression of\u00a0the\u00a0dimensionless quantity \u03bbw<\/sub>, on the energy conversion parameter between profiles\u00a01\u00a0and\u00a02, and also takes into account the conditions (Fr<\/em>1<\/sub>) at the inlet flow to the\u00a0transition zone of the channel, where the scour hole develops.<\/p>\n
\u03bbw<\/sub> <\/em>= 1.47-0.65ln (1-F<\/em>) Fr<\/em>1<\/sub> (R2<\/sup> = 0.978 \u2013 see Fig. 4<\/em> \u2013 Graph 2<\/em>)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (5)<\/p>\n
Experience from research into the shape a\u00a0scour hole, which is created by the effect of a\u00a0submerged horizontal jet on a\u00a0deformable bed [2], shows that the shape of a\u00a0scour hole at individual moments of its development is similar\u00a0\u2013 affine (Fig. 5<\/em>). An obvious similarity between the observed shape of the surface and the bed in the longitudinal profile can also be detected (Fig. 1 right<\/em>). Therefore, it is necessary to look for relationships between the basic geometric parameters of the scour hole (B2<\/sub>, B3<\/sub>, Lw<\/sub>, and Ys<\/sub>) and also between the levels of the water level (\u2206Hp<\/sub> = hp1<\/sub> – hp2<\/sub>) and the bed (Ys<\/sub>). Other derived relationships (Eq. 6, 7, 8<\/em>) drawn from experimentally obtained data (Fig. 4<\/em> \u2013 Graphs 3, 4, 5<\/em>) were again determined using statistical analysis [5, 6].<\/p>\n
B2<\/sub> <\/em>= 1.58Lw<\/sub><\/em> + 0.14\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (R2<\/sup> = 0.992 \u2013 see Fig. 4<\/em> \u2013 Graph 3<\/em>)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (6)<\/p>\n
B3<\/sub> <\/em>= 0.58B2<\/sub><\/em> + 0.10\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (R2<\/sup> = 0.978 \u2013 see Fig. 4<\/em> \u2013 Graph 4<\/em>)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (7)<\/p>\n
\u2206Hp<\/sub> = 0.51Ys<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (R2<\/sup> = 0.923 \u2013 see Fig. 4<\/em> \u2013 Graph 5<\/em>)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (8)<\/p>\n
The last statistically derived relationship from the experimentally obtained data on the \u201chalf\u201d model of the open channel can be seen in Fig. 4<\/em> \u2013 Graph 6<\/em>.<\/p>\n
<\/a>\n<\/p>\n
The author is aware of the fact that Eq. 9<\/em> can also have a\u00a0different form than the\u00a0one presented here, with the representation of a\u00a0number of other combinations of dimensionless quantities. According to [6], from the whole range of verified relationships between dimensionless quantities, the form presented above was selected based on the best achieved criterion value of the optimization process (R2<\/sup>) when including the minimum number of parameters intended for optimization (only the power exponent), which had a\u00a0positive effect on the\u00a0small error in their value. This equation mutually binds all three main parameters of scour hole geometry Lw<\/sub>, B2<\/sub>, Ys<\/sub> with the quantity B1 at the entrance to the\u00a0scour hole depending on the geomechanical properties of the soil (tg \u03c6 [-], d50<\/sub> [m]).<\/p>\n
In addition to the definition equation of the energy conversion parameter (Eq. 3a<\/em>) and the proposed empirical equations (Eq. 4\u20139<\/em>), in accordance with Fig.\u00a03<\/em>, the basic coupling equations for water stages (potential heights in balance profiles 1 and 2) can also be derived.<\/p>\n
hp<\/sub>1<\/sub> = Ys<\/sub> + H1<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10)<\/p>\n
hp<\/sub>2<\/sub> = Ys<\/sub> + H1<\/sub> – \u2206Hp<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (11)<\/p>\n
The last coupling equation serves to express the dimensionless length of the\u00a0wake \u03bbw<\/sub>, where Lw<\/sub> is the length of the wake corresponding approximately to the length of the scour hole.<\/p>\n
<\/a>\nAll the necessary quantitative relationships were derived for the design of the dimensions of the morphological object. All relationships are dimensionally homogeneous; they are presented in the form of dependence of the geometric dimensions of the scour hole (Eq. 6, 7, 8, 10, 11<\/em>) or dependence of dimensionless parameters (Eq. 3a, 4, 5, 9, 12<\/em>). Input data for these formulas are the main geometrical and hydraulic characteristics of the inlet flow (B1<\/sub>, H1<\/sub>, V1<\/sub>, Fr<\/em>1<\/sub>) in the section of the river channel before the opening of the stream and the geomechanical properties of the non-cohesive material in the open part (angle of repose of the soil under water \u03c6 and representative soil grain size d50<\/sub>). It should be noted that, in the case of applying experimentally derived equations, it is necessary to assume the validity of these relations for a\u00a0certain range of dimensionless parameters used. Dimensionless relations (Eq. 4, 5, 9<\/em>) were derived in the range of the\u00a0Fr<\/em>1<\/sub> parameter (0.75; 1.9) \u2013 i.e., rather in conditions characteristic of upland to mountain streams. The channel expansion ratio \u03b2 = B2<\/sub>\/B1<\/sub> in the scour hole in the experiments corresponded to the range (1.8; 3.5); i.e., flow\u00a0conditions with well to very well developed lateral vortices, which significantly participate in the formation of a\u00a03D scour hole with the characteristic \u201cpear\u201d shape. Dimensionless scour hole parameters were in this range: B2<\/sub>\/H1str<\/sub> (15; 50),<\/p>\n
Lw<\/sub>\/H1str<\/sub> (7; 23), and Ys<\/sub>\/H1str<\/sub> (0.6; 2.7). The energy conversion parameter F<\/em> was in the\u00a0range (0.04; 0.8). The experiments were conducted only on two types of granular non-cohesive bed, formed by significantly uniform-grained sharp-edged silica sands FP 1\u20131.6 and FP 1.6\u20134 mm with grain d50<\/sub> = 1.3 mm and 2.5\u00a0mm, and with inhomogeneity numbers U<\/em>\u00a0= d60<\/sub>\/d10<\/sub> 1.5 and 1.7, respectively. The specific weight of the sand was 2,516 kg\/m3<\/sup> and the volume weight 1,560 and 1,600 kg\/m3<\/sup> for both cases. The angle of repose under water was 33.8\u00b0 for finer-grained sand and 35.9\u00b0 for coarser sand.<\/p>\n
Changes of all relevant geometric dimensions, which are associated with the formation and development of a\u00a0scour hole, are dependent on the parameter, which is time \u2013 we can therefore speak of a\u00a0\u201cparametric scour hole\u201d [2]. In the case of designing a\u00a0\u201cpear\u201d scour hole as a\u00a0stabilized scour hole, it is necessary to start from one known dimension of the scour hole. In practice, a\u00a0frequent restriction for the design of a\u00a0\u201cpear\u201d scour hole is the limited width of the adjacent bank, where it is not necessary to solve complicated relationships with the owners of the surrounding land. Therefore, this limited dimension will be chosen, corresponding to the quantity B2<\/sub>; however, it is possible to choose any of the other dimensions (Ys<\/sub>, Lw<\/sub>). With this option, the time factor is indirectly introduced into the calculation. The designer of a\u00a0\u201cpear\u201d shaped scour hole does not need to know how much time t it would take to create a\u00a0scour hole of the calculated dimensions. What is essential is the fact that the dimension of the chosen scour hole (variable at time t) is a\u00a0limit beyond which it is no longer possible to go, and that the shape of the scour hole corresponds to one of the affine intermediate states (Fig. 5<\/em>), corresponding to the current extent of a\u00a0pair of vertical water holes. The \u201cpear\u201d scour hole must then be stabilized in the dimensions determined by the calculation, for example with embedded lined ribs in the bed and on the slopes at the beginning and end of the scour hole. However, specific technical measures for the stabilization of scour holes must be based on common local practice on the given watercourse, respecting in particular the character of the soils in the alluvial deposits and the available material base.<\/span><\/p>\n
<\/a>\nFig. 5. Parametric scour hole development \u2013 the scour hole shape for any time instant t is affine to the shape at another time instant t + 1<\/h6>\n
<\/a>\nFig. 6. Flow chart of design procedure of the 3D scour hole where one dimension is chosen: a) B2 is given; b) Ys is given; c) Lw is given. The referenced graphs are shown in Fig. 4<\/em>; the numbers of the referenced formulas correspond to the text<\/h6>\n
3D scour hole design methodology<\/h3>\n
The procedure for determining the basic dimensions of the scour hole in the\u00a0transition zone of the open river channel is based on Fig. 6<\/em>. For the inlet shape-fixed profile 1 of the lined river channel, the following flow characteristics are known: depth H1<\/sub>, velocity V1<\/sub>, value of the Froude number Fr<\/em>1<\/sub>, and width of the flow B1<\/sub> corresponding to the initial width of the channel in the transition area of the open channel. The design of the scour hole as a\u00a0spatial morphological element consists in determining its basic geometric dimensions B2<\/sub>, Lw<\/sub>, Ys<\/sub>, B3<\/sub>, and determining the reduced cross-sectional velocity V2<\/sub> in the extended profile 2 (Fig. 3<\/em>). The procedure for determining the basic dimensions of the scour hole differs for alternatives a), b) or c) in Fig. 6<\/em>, depending on whether the initial known or specified dimension of the scour hole is its maximum width B2<\/sub>, depth Ys<\/sub>, or length Lw<\/sub>. In all calculation alternatives, the channel widening ratio
\n\u03b2 = B2<\/sub>\/B1<\/sub> must first be determined; if B2<\/sub> is not known or specified, it must be estimated. Furthermore, it is necessary to determine the energy conversion parameter F<\/em> depending on \u03b2 and the value of Fr<\/em>1<\/sub> according to Graph 1<\/em> or from the\u00a0relationship (Eq. 4<\/em>) and according to Graph 2<\/em> or the dimensionless length of the\u00a0flood \u03bbw<\/sub> from the relationship (Eq. 5<\/em>), which connects the basic plan dimensions of the scour hole depression Lw<\/sub>, B1<\/sub> and B2<\/sub> through the relationship (Eq.\u00a012<\/em>). If B2<\/sub> was not known at the beginning of the calculation and it was necessary to choose it, its correct choice can be verified by the relationship (Eq. 12<\/em>), or also Graph 6<\/em>, or the relationship (
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