Correction methods for dropping of simulated water level utilising Preissmann and MOUSE slot models

Transcription

1 11 th International Conference on Urban Drainage, Edinburgh, Scotland, UK, 8 Correction methods for dropping of simulated water level utilising Preissmann and MOUSE slot models T. UKON 1, N. SHIGETA, M. WATANABE 3 *, H. SHIRAISHI 4 and M. UOTANI 5 1 Department of Design and Development, NIHON KOGYO Co., Ltd., 61, Kaniwa, Miki, Kita, KAGAWA, , JAPAN. Graduate School of Civil and Environmental Engineering, Ehime University, 3, Bunkyo-cho, Matsuyama, EHIME, , JAPAN. 3 Department of Civil and Environmental Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, EHIME, , JAPAN. 4 Aratani Civil Engineering Consultants Co., Ltd., -1-, Yogo-naka, Matsuyama, EHIME, 79-45, JAPAN. 5 ZERO Co., Ltd., 199-1, Takasu-cho, Onomichi, HIROSHIMA, , JAPAN. *Corresponding author, nabemasa@dpc.ehime-u.ac.jp ABSTRACT The Preissmann slot model and the MOUSE slot model have been widely used to simulate a pressurized flow in urban drainage sewer pipe systems. In the runoff simulations utilizing theses two models, the slot width is often widened to make the computational time increment large and save the computing time. The resulting simulated water level, however, is likely to decrease to the extent that the simulated water level can not be considered for practical use. The main reason for the tendency mentioned above, in the case of the Preissmann slot model, is that the discharge through the slot, the friction loss is not taken into account, increases considerably according to the enlargement of the slot width, and the main reason, in the case of the MOUSE slot model, is that the excess storage of runoff water in the slot increases to a considerable extent according to the enlargement. In this paper, firstly, two correction methods that can improve the accuracy of the runoff simulations are proposed for the Preissmann slot model and the MOUSE slot model respectively. Secondly, the adaptabilities of the methods are examined through numerical runoff experiments. KEYWORDS Preissmann slot model, MOUSE slot model, Lateral model, manholes, pressurized flow, urban drainage sewer pipe system INTRODUCTION Recently, a localized torrential rain has happened frequently by global warming and heat island, and the occurrence of the flood damage in urban drainage sewer pipe systems has increased. The countermeasures against the flood damage, therefore, must be planned as soon as possible. In order to achieve this, it is necessary the simulation model that can simulate precisely the storm water runoff with a pressurized flow in the sewer pipe system has been developed. The Preissmann and MOUSE slot models have been widely used to simulate accurately the pressurized flow in urban drainage sewer pipe systems. In the runoff simulations utilizing these two models, the slot width is often widened without an assurance that the accuracy of T. Ukon, et al. 1

2 11 th International Conference on Urban Drainage, Edinburgh, Scotland, UK, 8 the simulation results does not decrease to make the computational time increment large and save the computing time. The resulting simulated water level, however, is likely to decrease to the extent that the simulated water level can not be considered for practical use, especially in the case of runoff simulations with flooding from the sewer pipe systems. In this paper, two correction methods that can improve the accuracy of the runoff simulations are proposed for the Preissmann slot model and the MOUSE slot model respectively, and the adaptabilities of the methods are examined through numerical runoff experiments. PREISSMANN SLOT MODEL, MOUSE SLOT MODEL AND LATERAL MODEL Preissmann slot model In the Preissmann slot model, a very narrow slot is started up in the top of the sewer pipe as shown in Figure 1, and then an actual pressurized flow can be handled as an open-channel flow. The following assumptions are made in the derivation of the flow equations of the model. (1) Water is incompressible. () The sewer pipe is rigid. (3) The slot wall does not take part in friction loss. (4) The slot width is very narrow. (5) The slot width is decided depending on the sound wave velocity of the pressure wave. Using the assumptions mentioned above, the flow equations of motion and continuity for the pressurized flow are derived as follows (Cunge and Wegner, 1964) : 1 V V V n V V + + I + = (1) 4 / 3 g t g R c + V + t g V = ga gb ga S p c = = a 1 + ( y D) ; a = (3) BS Ap BS where V = the mean cross-sectional velocity; y = the depth of flow; R = the hydraulic radius; I = the slope of the sewer pipe; n = Manning s roughness coefficient; c = the celerity of the small gravity wave; B S = the slot width; A = the flow cross-sectional area; A p = the crosssectional area of the sewer pipe; D = the diameter of the sewer pipe; a = the pressure wave celerity; g = the gravitational acceleration; t = the time; x = the distance along the sewer pipe. () (Surcharged Flow) Hypothetical Slot Free-surface (Open-channel Flow) Figure 1. Preissmann hypothetical slot. Correction methods for dropping of simulated water level utilising Preissmann and MOUSE slot models

3 11 th International Conference on Urban Drainage, Edinburgh, Scotland, UK, 8 MOUSE slot model In the MOUSE slot model, a narrow slot is started up in the top of the sewer pipe as shown in Figure, and then an actual pressurized flow can be handled as an open-channel flow. The following assumptions are made in the derivation of the flow equations of the model. (1) Water is compressible. () The sewer pipe wall is elastic. (3) The slot part acts as not a flow area but a storage area. (4) The slot width is very narrow. Using the assumptions mentioned above, the flow equations of motion and continuity for pressurized flow are derived as follows (Danish Hydraulic Institute, 3): Q Q α A + + ga = ga( I I f ) (4) t Q Q ρ gap + + = (5) ρ a t 1+ g( y D) 1 + g( y D) Er e a ρ = ρ, A = Ap, ar =, a = (6) a ar ρ D 1+ a / ar where Q = the discharge; α = velocity distribution coefficient; I f = the friction slope; ρ = the density of water for a free surface flow; a = the speed of sound in water; E r = the Young s modulus of elasticity; e = the pipe wall thickness. B = B S = ga p /a B > B S D Figure. Pipe with a fictitious slot. Lateral model In urban drainage sewer pipe systems, many lateral pipes are installed to transport storm water from inlets to sewer pipes. Considering the storage of runoff water in the lateral pipes as shown in Figure 3, the momentum and continuity equations for the pressurized flow in a sewer pipe are obtained (Watanabe et al. 199, 1991). 1 V V V n V V + + I + = (7) 4 / 3 g t g R T. Ukon, et al. 3

4 11 th International Conference on Urban Drainage, Edinburgh, Scotland, UK, 8 c + t g V L N = (8) ( sinθ ) A A P c = g (9) l where c = the pressure wave celerity; L = the length of the sewer pipe; A l, N, and θ = the cross-sectional area, number, and angle of the lateral pipes, respectively. Street Inlet Ground Surface Laterals House Inlet Ground Surface y D Laterals θ Combined Sewer Pipe Combined Sewer Pipe Figure 3. Sewer pipe and lateral pipes. CORRECTION METHOD FOR DROPPING OF SIMULATED WATER LEVEL In the runoff simulations utilizing these two models, the slot width is often widened without an assurance that the accuracy of the simulation results does not decrease to make the computational time increment large and save the computing time. The resulting simulated water level, however, is likely to decrease to the extent that the simulated water level can not be considered for practical use. The purpose of this paper is to develop the correction methods for dropping of simulated water level using Preissmann slot model and MOUSE slot model. Correction method for Preissmann slot model First, a correction factor, Г, is introduced in the momentum equation (1) of the Preissmann slot model and the factor is multiplied by the term of the friction loss. Second, the steady pressurized flow condition is assumed and Г is derived so that the hydraulic grade line of the Preissmann slot model coincides with that of the Lateral model. ( y D) B V V S full Γ = F ± F A p c (1) c 4 / 3 R B S F = 1 + ( y D ) (11) 4 / 3 R A p where V full = the velocity of uniform pipe-full flow in the sewer pipe; R = the hydraulic radius except slot part; D = the length from invert to slot joint part; A p = the flow crosssectional area except slot part. 4 Correction methods for dropping of simulated water level utilising Preissmann and MOUSE slot models

5 11 th International Conference on Urban Drainage, Edinburgh, Scotland, UK, 8 The relation between the correction factor, Г, and y/d is shown in Figure 4. When the slot width is very narrow compared with the diameter of the sewer pipe (for example, B s =.1D), the simulated water level needs not being corrected because the discharge through the slot is very little compared with that through the sewer pipe. Therefore, Γ takes almost 1.. When the slot width is lager (for example, B s =.1D ~.5D), the discharge through the slot can not be ignored compared with that through the sewer pipe, and the simulated water level is likely to decrease because the slot wall does not have the friction loss. Therefore, in order to make the hydraulic grade line greater, Γ takes larger value than 1. according to the discharge through the slot. B S =.1D B S =.1D B S =.5D Γ y/d Figure 4. Correction factor, Г ~ y/d (D = 1. m). Correction method for MOUSE slot model The decrease of the simulated water level is due to the excess storage of runoff water, which is caused by widening the slot width, in the slot. In order to eliminate the excess storage, the same volume as the storage is reduced from the volume of the upper manhole as shown in Figure 5. The relation between the increment of the widened slot width and the reduced area of the manhole cross-sectional area is as follows: FM = α BS L (1) BS = B S BS (13) where FM = the reduced area of the manhole cross-sectional area; α = the correction coefficient (.5 α 1.); BS = the increment of the widened slot width; B S = the original slot width; B = the widened slot width. S V 1 = Excess storage of runoff water in slot Figure 5. Correction method for MOUSE slot model. V = Reduced volume of upper manhole V 1 = V T. Ukon, et al. 5

6 11 th International Conference on Urban Drainage, Edinburgh, Scotland, UK, 8 EXAMINATION OF ADDAPTABILITIES OF CORRECTION METHODS THROUGH NUMERICAL RUNOFF EXPERIMENTS The adaptabilities of the correction methods mentioned above were examined through the following numerical runoff experiments using the sewer pipe system as shown in Figure 6, Table 1, Table, and Table 3. The numerical runoff simulations under the following conditions were conducted: the inflow hydrographs at the upstream end of the pipe system are sinusoidal; the stage hydrographs at the downstream end are constant; the inundation on the ground surface never occurs; the periods of the inflow discharge hydrographs are 6 s and 1,8 s; the initial flow condition is the uniform pipe-full flow; the slot widths are assigned to be.1 m,.1 m, and.5 m. The simulated runoff hydrographs using the Preissmann slot model are shown in Figure 7 and Figure 8, and the ones using the MOUSE slot model are shown in Figure 9 and Figure 1. From these figures, it is apparent that the improvement of the dropping of the simulated water level as well as the accuracy of the simulated runoff hydrographs is achieved by making use of the correction methods. Q p Q Q full t T Level (m) 1 5 Q D Ground Surface L T G L T I Manholes Distance (m) Figure 6. Sewer pipe system used in numerical experiments. Table 1. Dimensions of sewer pipe system used in numerical experiments. D F M L T G I N T (m) (m N ) (m) (m) ( ) (m -1/3 Q s) p (s) Q full 1,8 Table. Slot width and pressure wave celerity. (D = 1. m) B S (m) a (m/s).1 D D D.5 Table 3. Corrected manhole cross-sectional area, F' M. (MOUSE slot model) α B' S =.1 m B' S =.5 m F M F' M F M F' M Correction methods for dropping of simulated water level utilising Preissmann and MOUSE slot models

7 11 th International Conference on Urban Drainage, Edinburgh, Scotland, UK, B S =.1m B S =.1m B S =.5m. Water Level (m) 7 6 Discharge (m 3 /s) (i) non- corrected (ii) corrected (i) non- corrected (ii) corrected (a) Water level hydrographs (b) Discharge hydrographs Figure 7. Simulated hydrographs (Preissmann slot model, T = 6 s). 9 8 B S =.1m B S =.1m B S =.5m. Water Level (m) 7 6 Discharge (m 3 /s) (i) non- corrected (ii) corrected (i) non- corrected (ii) corrected (a) Water level hydrographs (b) Discharge hydrographs Figure 8. Simulated hydrographs (Preissmann slot model, T = 1,8 s). T. Ukon, et al. 7

8 11 th International Conference on Urban Drainage, Edinburgh, Scotland, UK, B S =.1m B S =.1m B S =.5m. α =.75 α =.75 Water Level (m) 7 6 Discharge (m 3 /s) (i) non- corrected (ii) corrected (i) non- corrected (ii) corrected (a) Water level hydrographs (b) Discharge hydrographs Figure 9. Simulated hydrographs (MOUSE slot mode, T = 6 s). 9 8 B S =.1m B S =.1m B S =.5m. α =.75 α =.75 Water Level (m) 7 6 Discharge (m 3 /s) (i) non- corrected (ii) corrected (i) non- corrected (ii) corrected (a) Water level hydrographs (b) Discharge hydrographs Figure 1. Simulated hydrographs (MOUSE slot model, T = 1,8 s). 8 Correction methods for dropping of simulated water level utilising Preissmann and MOUSE slot models

9 11 th International Conference on Urban Drainage, Edinburgh, Scotland, UK, 8 CONCLUSIONS It was clarified that the dropping of the simulated water level caused by widening the slot width can be improved through the simulations making use of the correction methods proposed in this paper. REFERENCES Cunge, J. A. and Wegner, M. (1964). Numerical integration of Barré de Saint-Venant s flow equations by means of an implicit scheme of finite differences. La Houille Blanche., No.1, Chaudhry, M. H. (1979). Applied Hydraulic Transients. VNR Company, New York, Watanabe, M., Etoh, T. and Murota, A. (199). Runoff simulation model of sewer pipe systems considering pressure-relaxation effect by lateral pipes. Proc. 5 th Int. Conf. on Urban Storm Drainage..1, Watanabe, M., Etoh, T. and Murota, A. (1991). Runoff simulation of sewer pipe systems with lateral pipes. Journal of Natural Disaster Science., 13(1), MOUSE PIPE FLOW Reference Manual. (3). Danish Hydraulic Institute, Denmark, Chapter 3, 1-7. T. Ukon, et al. 9