• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.33-41
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• 1996

In this article, all the universal totally positive integral ternary lattices over $Q(\sqrt{5})$ are determined.

• Magazine of the Korean Society of Agricultural Engineers
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• v.20 no.1
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• pp.4592-4598
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• 1978

The purpose of this research is to find out mathemetically an economical intake water depth in the design of head works through the derivation of some formulas. For the performance of the purpose the following formulas were found out for the design intake water depth in each flow type of intake sluice, such as overflow type and orifice type. (1) The conditional equations of !he economical intake water depth in .case that weir body is placed on permeable soil layer ; (a) in the overflow type of intake sluice, {{{{ { zp}_{1 } { Lh}_{1 }+ { 1} over {2 } { Cp}_{3 }L(0.67 SQRT { q} -0.61) { ( { d}_{0 }+ { h}_{1 }+ { h}_{0 } )}^{- { 1} over {2 } }- { { { 3Q}_{1 } { p}_{5 } { h}_{1 } }^{- { 5} over {2 } } } over { { 2m}_{1 }(1-s) SQRT { 2gs} }+[ LEFT { b+ { 4C TIMES { 0.61}^{2 } } over {3(r-1) }+z( { d}_{0 }+ { h}_{0 } ) RIGHT } { p}_{1 }L+(1+ SQRT { 1+ { z}^{2 } } ) { p}_{2 }L+ { dcp}_{3 }L+ { nkp}_{5 }+( { 2z}_{0 }+m )(1-s) { L}_{d } { p}_{7 } ] =0}}}} (b) in the orifice type of intake sluice, {{{{ { zp}_{1 } { Lh}_{1 }+ { 1} over {2 } C { p}_{3 }L(0.67 SQRT { q} -0.61)}}}} {{{{ { ({d }_{0 }+ { h}_{1 }+ { h}_{0 } )}^{ - { 1} over {2 } }- { { 3Q}_{1 } { p}_{ 6} { { h}_{1 } }^{- { 5} over {2 } } } over { { 2m}_{ 2}m' SQRT { 2gs} }+[ LEFT { b+ { 4C TIMES { 0.61}^{2 } } over {3(r-1) }+z( { d}_{0 }+ { h}_{0 } ) RIGHT } { p}_{1 }L }}}} {{{{+(1+ SQRT { 1+ { z}^{2 } } ) { p}_{2 } L+dC { p}_{4 }L+(2 { z}_{0 }+m )(1-s) { L}_{d } { p}_{7 }]=0 }}}} where, z=outer slope of weir body (value of cotangent), h1=intake water depth (m), L=total length of weir (m), C=Bligh's creep ratio, q=flood discharge overflowing weir crest per unit length of weir (m3/sec/m), d0=average height to intake sill elevation in weir (m), h0=freeboard of weir (m), Q1=design irrigation requirements (m3/sec), m1=coefficient of head loss (0.9∼0.95) s=(h1-h2)/h1, h2=flow water depth outside intake sluice gate (m), b=width of weir crest (m), r=specific weight of weir materials, d=depth of cutting along seepage length under the weir (m), n=number of side contraction, k=coefficient of side contraction loss (0.02∼0.04), m2=coefficient of discharge (0.7∼0.9) m'=h0/h1, h0=open height of gate (m), p1 and p4=unit price of weir body and of excavation of weir site, respectively (won/㎥), p2 and p3=unit price of construction form and of revetment for protection of downstream riverbed, respectively (won/㎡), p5 and p6=average cost per unit width of intake sluice including cost of intake canal having the same one as width of the sluice in case of overflow type and orifice type respectively (won/m), zo : inner slope of section area in intake canal from its beginning point to its changing point to ordinary flow section, m: coefficient concerning the mean width of intak canal site,a : freeboard of intake canal. (2) The conditional equations of the economical intake water depth in case that weir body is built on the foundation of rock bed ; (a) in the overflow type of intake sluice, {{{{ { zp}_{1 } { Lh}_{1 }- { { { 3Q}_{1 } { p}_{5 } { h}_{1 } }^{- {5 } over {2 } } } over { { 2m}_{1 }(1-s) SQRT { 2gs} }+[ LEFT { b+z( { d}_{0 }+ { h}_{0 } )RIGHT } { p}_{1 }L+(1+ SQRT { 1+ { z}^{2 } } ) { p}_{2 }L+ { nkp}_{5 }}}}} {{{{+( { 2z}_{0 }+m )(1-s) { L}_{d } { p}_{7 } ]=0 }}}} (b) in the orifice type of intake sluice, {{{{ { zp}_{1 } { Lh}_{1 }- { { { 3Q}_{1 } { p}_{6 } { h}_{1 } }^{- {5 } over {2 } } } over { { 2m}_{2 }m' SQRT { 2gs} }+[ LEFT { b+z( { d}_{0 }+ { h}_{0 } )RIGHT } { p}_{1 }L+(1+ SQRT { 1+ { z}^{2 } } ) { p}_{2 }L}}}} {{{{+( { 2z}_{0 }+m )(1-s) { L}_{d } { p}_{7 } ]=0}}}} The construction cost of weir cut-off and revetment on outside slope of leeve, and the damages suffered from inundation in upstream area were not included in the process of deriving the above conditional equations, but it is true that magnitude of intake water depth influences somewhat on the cost and damages. Therefore, in applying the above equations the fact that should not be over looked is that the design value of intake water depth to be adopted should not be more largely determined than the value of h1 satisfying the above formulas.

• Magazine of the Korean Society of Agricultural Engineers
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• v.17 no.1
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• pp.3642-3654
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• 1975

The purpose of this thesis is to derive a unit hydrograph which may be applied to the ungaged watershed area from the relations between directly measurable unitgraph properties such as peak discharge(qp), time to peak discharge (Tp), and lag time (Lg) and watershed characteristics such as river length(L) from the given station to the upstream limits of the watershed area in km, river length from station to centroid of gravity of the watershed area in km (Lca), and main stream slope in meter per km (S). Other procedure based on routing a time-area diagram through catchment storage named Instantaneous Unit Hydrograph(IUH). Dimensionless unitgraph also analysed in brief. The basic data (1969 to 1973) used in these studies are 9 recording level gages and rating curves, 41 rain gages and pluviographs, and 40 observed unitgraphs through the 9 sub watersheds in Nak Oong River basin. The results summarized in these studies are as follows; 1. Time in hour from start of rise to peak rate (Tp) generally occured at the position of 0.3Tb (time base of hydrograph) with some indication of higher values for larger watershed. The base flow is comparelatively higher than the other small watershed area. 2. Te losses from rainfall were divided into initial loss and continuing loss. Initial loss may be defined as that portion of storm rainfall which is intercepted by vegetation, held in deppression storage or infiltrated at a high rate early in the storm and continuing loss is defined as the loss which continues at a constant rate throughout the duration of the storm after the initial loss has been satisfied. Tis continuing loss approximates the nearly constant rate of infiltration (${\Phi}$-index method). The loss rate from this analysis was estimated 50 Per cent to the rainfall excess approximately during the surface runoff occured. 3. Stream slope seems approximate, as is usual, to consider the mainstreamonly, not giving any specific consideration to tributary. It is desirable to develop a single measure of slope that is representative of the who1e stream. The mean slope of channel increment in 1 meter per 200 meters and 1 meter per 1400 meters were defined at Gazang and Jindong respectively. It is considered that the slopes are low slightly in the light of other river studies. Flood concentration rate might slightly be low in the Nak Dong river basin. 4. It found that the watershed lag (Lg, hrs) could be expressed by Lg=0.253 (L.Lca)0.4171 The product L.Lca is a measure of the size and shape of the watershed. For the logarithms, the correlation coefficient for Lg was 0.97 which defined that Lg is closely related with the watershed characteristics, L and Lca. 5. Expression for basin might be expected to take form containing theslope as {{{{ { L}_{g }=0.545 {( { L. { L}_{ca } } over { SQRT {s} } ) }^{0.346 } }}}} For the logarithms, the correlation coefficient for Lg was 0.97 which defined that Lg is closely related with the basin characteristics too. It should be needed to take care of analysis which relating to the mean slopes 6. Peak discharge per unit area of unitgraph for standard duration tr, ㎥/sec/$\textrm{km}^2$, was given by qp=10-0.52-0.0184Lg with a indication of lower values for watershed contrary to the higher lag time. For the logarithms, the correlation coefficient qp was 0.998 which defined high sign ificance. The peak discharge of the unitgraph for an area could therefore be expected to take the from Qp=qp. A(㎥/sec). 7. Using the unitgraph parameter Lg, the base length of the unitgraph, in days, was adopted as {{{{ {T}_{b } =0.73+2.073( { { L}_{g } } over {24 } )}}}} with high significant correlation coefficient, 0.92. The constant of the above equation are fixed by the procedure used to separate base flow from direct runoff. 8. The width W75 of the unitgraph at discharge equal to 75 per cent of the peak discharge, in hours and the width W50 at discharge equal to 50 Per cent of the peak discharge in hours, can be estimated from {{{{ { W}_{75 }= { 1.61} over { { q}_{b } ^{1.05 } } }}}} and {{{{ { W}_{50 }= { 2.5} over { { q}_{b } ^{1.05 } } }}}} respectively. This provides supplementary guide for sketching the unitgraph. 9. Above equations define the three factors necessary to construct the unitgraph for duration tr. For the duration tR, the lag is LgR=Lg+0.2(tR-tr) and this modified lag, LgRis used in qp and Tb It the tr happens to be equal to or close to tR, further assume qpR=qp. 10. Triangular hydrograph is a dimensionless unitgraph prepared from the 40 unitgraphs. The equation is shown as {{{{ { q}_{p } = { K.A.Q} over { { T}_{p } } }}}} or {{{{ { q}_{p } = { 0.21A.Q} over { { T}_{p } } }}}} The constant 0.21 is defined to Nak Dong River basin. 11. The base length of the time-area diagram for the IUH routing is {{{{C=0.9 {( { L. { L}_{ca } } over { SQRT { s} } ) }^{1/3 } }}}}. Correlation coefficient for C was 0.983 which defined a high significance. The base length of the T-AD was set to equal the time from the midpoint of rain fall excess to the point of contraflexure. The constant K, derived in this studies is K=8.32+0.0213 {{{{ { L} over { SQRT { s} } }}}} with correlation coefficient, 0.964. 12. In the light of the results analysed in these studies, average errors in the peak discharge of the Synthetic unitgraph, Triangular unitgraph, and IUH were estimated as 2.2, 7.7 and 6.4 per cent respectively to the peak of observed average unitgraph. Each ordinate of the Synthetic unitgraph was approached closely to the observed one.

• Magazine of the Korean Society of Agricultural Engineers
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• v.19 no.1
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• pp.4296-4311
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• 1977

This study was conducted to derive an Instantaneous Unit Hydrograph for the accurate and reliable unitgraph which can be used to the estimation and control of flood for the development of agricultural water resources and rational design of hydraulic structures. Eight small watersheds were selected as studying basins from Han, Geum, Nakdong, Yeongsan and Inchon River systems which may be considered as a main river systems in Korea. The area of small watersheds are within the range of 85 to 470$\textrm{km}^2$. It is to derive an accurate Instantaneous Unit Hydrograph under the condition of having a short duration of heavy rain and uniform rainfall intensity with the basic and reliable data of rainfall records, pluviographs, records of river stages and of the main river systems mentioned above. Investigation was carried out for the relations between measurable unitgraph and watershed characteristics such as watershed area, A, river length L, and centroid distance of the watershed area, Lca. Especially, this study laid emphasis on the derivation and application of Instantaneous Unit Hydrograph (IUH) by applying Nash's conceptual model and by using an electronic computer. I U H by Nash's conceptual model and I U H by flood routing which can be applied to the ungaged small watersheds were derived and compared with each other to the observed unitgraph. 1 U H for each small watersheds can be solved by using an electronic computer. The results summarized for these studies are as follows; 1. Distribution of uniform rainfall intensity appears in the analysis for the temporal rainfall pattern of selected heavy rainfall event. 2. Mean value of recession constants, Kl, is 0.931 in all watersheds observed. 3. Time to peak discharge, Tp, occurs at the position of 0.02 Tb, base length of hlrdrograph with an indication of lower value than that in larger watersheds. 4. Peak discharge, Qp, in relation to the watershed area, A, and effective rainfall, R, is found to be {{{{ { Q}_{ p} = { 0.895} over { { A}^{0.145 } } }}}} AR having high significance of correlation coefficient, 0.927, between peak discharge, Qp, and effective rainfall, R. Design chart for the peak discharge (refer to Fig. 15) with watershed area and effective rainfall was established by the author. 5. The mean slopes of main streams within the range of 1.46 meters per kilometer to 13.6 meter per kilometer. These indicate higher slopes in the small watersheds than those in larger watersheds. Lengths of main streams are within the range of 9.4 kilometer to 41.75 kilometer, which can be regarded as a short distance. It is remarkable thing that the time of flood concentration was more rapid in the small watersheds than that in the other larger watersheds. 6. Length of main stream, L, in relation to the watershed area, A, is found to be L=2.044A0.48 having a high significance of correlation coefficient, 0.968. 7. Watershed lag, Lg, in hrs in relation to the watershed area, A, and length of main stream, L, was derived as Lg=3.228 A0.904 L-1.293 with a high significance. On the other hand, It was found that watershed lag, Lg, could also be expressed as {{{{Lg=0.247 { ( { LLca} over { SQRT { S} } )}^{ 0.604} }}}} in connection with the product of main stream length and the centroid length of the basin of the watershed area, LLca which could be expressed as a measure of the shape and the size of the watershed with the slopes except watershed area, A. But the latter showed a lower correlation than that of the former in the significance test. Therefore, it can be concluded that watershed lag, Lg, is more closely related with the such watersheds characteristics as watershed area and length of main stream in the small watersheds. Empirical formula for the peak discharge per unit area, qp, ㎥/sec/$\textrm{km}^2$, was derived as qp=10-0.389-0.0424Lg with a high significance, r=0.91. This indicates that the peak discharge per unit area of the unitgraph is in inverse proportion to the watershed lag time. 8. The base length of the unitgraph, Tb, in connection with the watershed lag, Lg, was extra.essed as {{{{ { T}_{ b} =1.14+0.564( { Lg} over {24 } )}}}} which has defined with a high significance. 9. For the derivation of IUH by applying linear conceptual model, the storage constant, K, with the length of main stream, L, and slopes, S, was adopted as {{{{K=0.1197( {L } over { SQRT {S } } )}}}} with a highly significant correlation coefficient, 0.90. Gamma function argument, N, derived with such watershed characteristics as watershed area, A, river length, L, centroid distance of the basin of the watershed area, Lca, and slopes, S, was found to be N=49.2 A1.481L-2.202 Lca-1.297 S-0.112 with a high significance having the F value, 4.83, through analysis of variance. 10. According to the linear conceptual model, Formular established in relation to the time distribution, Peak discharge and time to peak discharge for instantaneous Unit Hydrograph when unit effective rainfall of unitgraph and dimension of watershed area are applied as 10mm, and $\textrm{km}^2$ respectively are as follows; Time distribution of IUH {{{{u(0, t)= { 2.78A} over {K GAMMA (N) } { e}^{-t/k } { (t.K)}^{N-1 } }}}} (㎥/sec) Peak discharge of IUH {{{{ {u(0, t) }_{max } = { 2.78A} over {K GAMMA (N) } { e}^{-(N-1) } { (N-1)}^{N-1 } }}}} (㎥/sec) Time to peak discharge of IUH tp=(N-1)K (hrs) 11. Through mathematical analysis in the recession curve of Hydrograph, It was confirmed that empirical formula of Gamma function argument, N, had connection with recession constant, Kl, peak discharge, QP, and time to peak discharge, tp, as {{{{{ K'} over { { t}_{ p} } = { 1} over {N-1 } - { ln { t} over { { t}_{p } } } over {ln { Q} over { { Q}_{p } } } }}}} where {{{{K'= { 1} over { { lnK}_{1 } } }}}} 12. Linking the two, empirical formulars for storage constant, K, and Gamma function argument, N, into closer relations with each other, derivation of unit hydrograph for the ungaged small watersheds can be established by having formulars for the time distribution and peak discharge of IUH as follows. Time distribution of IUH u(0, t)=23.2 A L-1S1/2 F(N, K, t) (㎥/sec) where {{{{F(N, K, t)= { { e}^{-t/k } { (t/K)}^{N-1 } } over { GAMMA (N) } }}}} Peak discharge of IUH) u(0, t)max=23.2 A L-1S1/2 F(N) (㎥/sec) where {{{{F(N)= { { e}^{-(N-1) } { (N-1)}^{N-1 } } over { GAMMA (N) } }}}} 13. The base length of the Time-Area Diagram for the IUH was given by {{{{C=0.778 { ( { LLca} over { SQRT { S} } )}^{0.423 } }}}} with correlation coefficient, 0.85, which has an indication of the relations to the length of main stream, L, centroid distance of the basin of the watershed area, Lca, and slopes, S. 14. Relative errors in the peak discharge of the IUH by using linear conceptual model and IUH by routing showed to be 2.5 and 16.9 percent respectively to the peak of observed unitgraph. Therefore, it confirmed that the accuracy of IUH using linear conceptual model was approaching more closely to the observed unitgraph than that of the flood routing in the small watersheds.