The previous research in the area of floating debris jams focused on the following issues:

- Floating debris jams at natural obstacles in rivers ( [1] [2] [3] ),

- The influence of different layouts of wooden debris retention racks on the resulting jams ( [4] [5] [6] [7] ),

- The probability of debris jams at bridges ( [8] [9] ) and spillways ( [10] [11] [12] [13] ).

Only little research exists in the area of the consequences of floating debris jams at spillways and wooden debris retention racks, in particular the resulting backwater effect, which is the focus of this paper.

The approach of this study envisaged a dimensional analysis of the backwater effect caused by floating debris jams. For the detected parameters, the Froude number of the approach flow and the debris density, the results of different hydraulic model test series concerning debris jams at spillways as well as at racks for the retention of wooden debris were uniformly evaluated.

The backwater effect caused by floating debris jams directly depends on the development and shape of the jam. The dimensional analysis of the development and shape of debris jams as well as the resulting backwater effect can be performed with only one characteristic parameter ( [14] [15] ). The characteristic parameter selected in this study is the so-called debris jam compactness. The debris jam compactness is the ratio of height T to length L of the debris jam; i.e. compactness is T/L. T and L are illustrated in the example of a debris jam at a spillway bay in Figure 1 and at a debris retention rack in Figure 2.

A number of different variables likely have an influence on the debris jam

compactness T/L. First, a higher approach flow velocity v in the initial situation prior to debris jam formation probably leads to a higher compactness. Different overflow heights or flow depths h should also affect compactness. Gravitational acceleration g together with the densities of the debris ρ_D and water ρ_W can cause some logs to plunge below others floating on the water surface. The compactness T/L of debris jams is thus regarded as a function of v, g, h, ρ_D and ρ_W. Specifically, it will be treated as a product of a proportionality constant C_T/L and of v, g, h, ρ_D and ρ_W with the different exponents a, b, c, d and e:

T / L = f ( v , g , h , ρ D , ρ W ) = C T / L ⋅ v a ⋅ g b ⋅ h c ⋅ ρ D d ⋅ ρ W e (1)

The dimensions matrix A_T/L in the mass[m]-length[l]-time[t]-system takes the following form:

A T / L = [ v g h ρ D ρ W m 0 0 0 1 1 l 1 1 1 − 3 − 3 t − 1 − 2 0 0 0 ] (2)

Because the compactness T/L is dimensionless, solving the dimensions equations of the mass, the length and the time leads to the following:

[ m ] : 0 = d + e ⇒ e = − d (3)

[ l ] : 0 = a + b + c − 3 ⋅ d − 3 ⋅ e ⇒ c = − a − b = − 0.5 ⋅ a (4)

[ t ] : 0 = − a − 2 ⋅ b ⇒ b = − 0.5 ⋅ a (5)

⇒ T / L = C T / L ⋅ v a ⋅ g b ⋅ h c ⋅ ρ D d ⋅ ρ W e = C T / L ⋅ v a ⋅ g − 0.5 a ⋅ h − 0.5 a ⋅ ρ D d ⋅ ρ W − d                         = C T / L ⋅ [ v / ( g ⋅ h ) 0.5 ] a ⋅ ( ρ D / ρ W ) d = C T / L ⋅ F 0 a ⋅ ( ρ D / ρ W ) d (6)

The compactness of the debris jam T/L is a product of the proportionality constant C_T/L, the Froude number F₀ of the approach flow prior to debris jam formation with the unknown exponent a and of the density ratio of debris and water ρ_D/ρ_W (relative debris density) with the unknown exponent d. This same fundamental dependence is found for the dimensionless parameter relative backwater effect Δh/h caused by a debris jam, however with different number values for C_T/L, a and d.

It was the aim to detect the influence of the Froude number F₀ and the relative debris density ρ_D/ρ_W on the backwater effect Δh/h caused by debris jams at spillways more precisely. Therefore, the results of a large-scale hydraulic model test series at the Laboratory of Hydraulic and Water Resources Engineering of the Technical University of Munich was evaluated [16] . The model of a spillway intake structure (Figure 3) was built in a rectangular canal on the open-air area of the laboratory (canal length: 220 m, width: 2.5 m, depth: 2 m). The scale of the Froude model was 1:20. The intake structure consisted of three identical bays

with a width W = 50 cm, WES-shaped crests and radial gates. The gates were fully opened during the described tests and without any influence on the jam process. The standardised dimensions of the model intake structure were representative for many existing structures. With regard to the debris, 100-logs sets with the distribution of the log length l_L shown in Table 1 and model log diameters d_L between 2 cm and 4 cm were used. The logs had no or only a few branches. Describing the procedure of a single test an initial steady-state flow condition without a debris jam was established in the canal first. Then groups of five logs were added to the flow upstream of the spillway intake structure until all three bays were blocked and all remaining logs of the set were added to the flow. The geometric development of the debris jam finally consisting of the total log set was observed. After reaching a new steady-state flow condition, the upstream water level was measured and the relative backwater effect Δh/h was derived as the main result.

In the hydraulic model tests concerning debris jams at spillways the relative debris density ρ_D/ρ_W was varied for a more precise detection of its influence on the relative backwater effect Δh/h. To avoid density fluctuations which occur in natural wood due to shrinkage and expansion, artificial 100-logs sets with four different densities (ρ_D/ρ_W = 0.8; 0.9; 0.95 and 0.975) were used. Aside from these density variations, the debris mixes were identical. The quantitative influence of the debris density on the development of the jam and on the backwater effect is comparable to the Froude number of the approach flow. With a rising relative debris density ρ_D/ρ_W the debris jams became more compact and the relative backwater effect Δh/h increased. This means that the exponent d in (6) is positive. Figure 7 shows a loose, single-layer floating carpet for ρ_D/ρ_W = 0.8 (left) and a compact multi-layer debris body for ρ_D/ρ_W = 0.975 (right). In both cases the Froude number was F₀ = 0.08. The steel grid in front of the spillway visible in Figure 7 was required to prevent some logs of the artificial mix from passing over the spillway.

Figure 8 shows the measured value pairs of the relative debris density ρ_D/ρ_W and the relative backwater effect Δh/h for two different Froude numbers of the approach flow. For F₀ = 0.08, the relative backwater effect increases from Δh/h = 5.9% for the smallest relative debris density ρ_D/ρ_W = 0.8 up to Δh/h = 45.9% for the largest relative debris density ρ_D/ρ_W = 0.975. For the higher Froude number F₀ = 0.14, the respective values of the relative backwater effect are larger, which corresponds with the findings described in 4.1. In this test series, the relative backwater effect increases from Δh/h = 15.7% for ρ_D/ρ_W = 0.8 up to Δh/h = 46.4% for ρ_D/ρ_W = 0.975.

Similar to the dependence on the Froude number of the approach flow, a more exact quantification of the dependence of the backwater effect on the relative debris density in the form of the exponent d in (6) is not considered useful

for the present test results. The reason is the randomness of the debris jam development which might possibly be eliminated from the test results if the shapes of the debris jams are default in further test series.

Current and previous hydraulic model tests at the Laboratory of Hydraulic and Water Resources Engineering of the Technical University of Munich concerning debris jams at spillways as well as at racks for the retention of wooden debris were uniformly evaluated. The evaluation resulted in the Froude number of the approach flow and the debris density as the decisive parameters. The higher the Froude number of the approach flow and the larger the debris density, the more compact is the debris jam and the higher is its backwater effect. Due to the randomness of the debris jam development a more exact quantification of the dependence of the backwater effect on the Froude number of the approach flow and on the debris density is not considered useful for the present results.

The hydraulic model tests concerning debris jams at spillways performed in the course of this research were funded by the Swedish company ELFORSK AB. Many thanks for the support and cooperation.

Hartlieb, A. (2017) Decisive Parameters for Backwater Effects Caused by Floating Debris Jams. Open Journal of Fluid Dynamics, 7, 475-484. https://doi.org/10.4236/ojfd.2017.74032

The following symbols are used in this paper:

a = exponent

A_T/L = dimensions matrix

b = exponent

c = exponent

C_T/L = proportionality constant

d = exponent

d_L = log diameter

e = exponent

F₀ = Froude number of the approach flow prior to debris jam formation

g = gravitational acceleration

h = flow height or flow depth

l = length

l_L = log length

L = length of the debris jam

m = mass

t = time

T = height of the debris jam

T/L = debris jam compactness

v = approach flow velocity prior to debris jam formation

W = width of spillway bay

Δh/h= relative backwater effect

ρ_D = density of the debris

ρ_D/ρ_W= relative debris density

ρ_W = density of water