_{1}

The dimensional analysis of the backwater effect caused by debris jams results in the Froude number of the approach flow in the initial situation prior to debris jam formation and the debris density as decisive parameters. For the more precise detection of the influence of both parameters the results of different hydraulic model test series 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. On the one hand a significant increase of the backwater effect with a rising Froude number of the approach flow could be shown. This is in good correlation to recent test results for debris jams at retention racks at the Laboratory of Hydraulics, Hydrology and Glaciology of the Swiss Federal Institute of Technology Zurich. On the other hand a significant increase of the backwater effect could also be shown for a rising debris density. However, the test results also show that significantly different backwater effects can occur in different test runs with identical test conditions. These differences are a result of the randomness of debris jam development, and therefore, 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 previous research in the area of floating debris jams focused on the following issues:

- Floating debris jams at natural obstacles in rivers ( [

- The influence of different layouts of wooden debris retention racks on the resulting jams ( [

- The probability of debris jams at bridges ( [

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 ( [

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 F_{0} 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 F_{0} 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 [

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 _{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.

The test results were evaluated with the focus on the influence of the Froude number F_{0} of the approach flow prior to debris jam formation on the relative backwater effect Δh/h. For this evaluation only the results of tests with natural debris and a mean relative debris density of ρ_{D}/ρ_{W} = 0.8 were used. For higher Froude numbers F_{0} > 0.30, multi-layer debris bodies with high compactness T/L (see _{0} < 0.15, loose single-layer floating carpets with low compactness T/L and correspondingly low relative backwater effects Δh/h < 6% (see

l_{L}/W | 1.0 | 1.3 | 1.5 | 1.7 | 2.0 |
---|---|---|---|---|---|

Number of logs | 15 | 20 | 30 | 20 | 15 |

determined for the same 100-logs set of debris. From these results, it can be deduced that the relative backwater effect Δh/h increases with rising Froude number F_{0} of the approach flow. This means that the exponent a in (6) is positive. But the test results in

Knauss has performed fundamental tests for debris jams at retention racks with different layouts of vertical pillars [_{0} of the approach flow (see _{0} = 1.61 a relative backwater effect Δh/h = 220% was determined. And the highest Froude number F_{0} = 2.45 caused a relative backwater effect Δh/h = 430%. For comparison, the correlations derived from the test results of Weitbrecht and Schmocker [

Two value pairs for F_{0} and Δh/h can be adopted from Weitbrecht and Schmocker [_{0} < 1.5. Schmocker and Hager repeated each test with identical conditions twice and obtained clearly divergent backwater effects due to the randomness of the debris jam development, too. By the calculation of the mean values of the backwater effect for each Froude number

they derived the linear correlation. This could rather be seen critically, because the possible strong variation of the backwater effect for identical conditions is lost in the consideration. Using the parameters defined in this paper, the equation of Schmocker and Hager [_{0} is:

( Δ h + h ) / h = 1.4 + 1.9 ⋅ F 0 or Δ h / h = 0.4 + 1.9 ⋅ F 0 (7)

The quantity of the increase of the relative backwater effect Δh/h with the rising Froude number F_{0} lies in a comparable range in all test series. If the linear correlation of Schmocker und Hager [

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. _{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 F_{0} = 0.08. The steel grid in front of the spillway visible in

_{D}/ρ_{W} and the relative backwater effect Δh/h for two different Froude numbers of the approach flow. For F_{0} = 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 F_{0} = 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

F_{0} = 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