【導(dǎo)讀】research.E-mailaddress:(S.Arthur).

　　

【正文】 d pressurehead (V and H, respectively). These equations may thenbe solved using the method of characteristics [7–9]. Usingthis method, the pipeline is divided into N sections (N+1Fig. 9. Details of the normal application of the method of characteristics.(Note: For clarity only alternate nodes are represented.)nodes) of equal length, SOHx. Values of discharge and headare known at each node at t=0, as detailed above. Thesolution framework normally employed to solve the hydraulic conditions present in the horizontal pipework isillustrated in Fig. 9, where points A and B represent twopoints in space and time (nodes i?1 and i+1att=0)where discharge and pressure head are known, and pointP represents the intermediate node, i,att=SOHt. The nextstep in the calculation procedure is to determine the Qand H at each calculation node at t =SOHt (SOHt is determined using Eq. (4) — the Courant Criterion), this isdone by municating the hydraulic conditions from adjacent nodes for the previous time step to the calculationpoint. This is acplished by applying the characteristicEquations (Eqs. (5) and (6) which are valid along C+andC?, respectively) and intersect at point P:V@V@x+@V@t+g@H@x+fV|V|2R=0。 (2)S. Arthur, . SwaSOeld/Building and Environment 36 (2020) 939–948 947SUBc2@V@x+VSUBgparenleftbigg@H@x+sinVTparenrightbigg+SUBg@H@t=0。 (3)SOHt6SOHxV +c～=SOHxc(when cp14V) (4)Vp=VA?g(Hp?HA)cA+gsinVTASOHt?fAVA|VA|SOHt2RA。 (5)Vp=VB+g(Hp?HB)cB+gsinVTBSOHt?fBVB|VB|SOHt2RB: (6)At each end of the pipe length considered in Fig. 9only one of the characteristic equations is available (.at the upstream end only the C?characteristic, and at thedownstream end only the C+characteristic). Therefore, fora solution to be reached at these points, additional relationships must be formulated which represent Q and H atthe upstream and downstream boundaries. The system exitboundary consists of setting the pressure to atmospheric atthe point of exit. Whilst the entry relies on an empiricalrelationship which relates CRow depth to outlet type.Any CRow entering the system during the priming phaseafter full bore CRow has been established is assumed tocontain % air as the roof outlet is fully submerged.The CRow downstream of the jump is assumed to be a homogeneous air=water mixture between adjacent nodes. Thepropagation velocities between internodal reaches may nowbe puted using Eq. (7).1It can be seen that the propagation velocity will not be equal throughout the system,and that the CRow velocity may also approach the propagation velocity under some conditions. For the horizontalpipe length, this consideration was found not to be important. However, as the air content of the CRow signiFFcantlyinCRuences the ambient pressures within the vertical stack,the inCRuence that the air content has on the propagationvelocity here must also be taken into account. This, therefore, results in a variation in wave speed between theponent pipe lengths within the system and betweeninternodal sections. Therefore, if SOHt is selected using thehighest wave speed, sections of the system may exist inwhich the SOHt time step is signiFFcantly smaller than thatprescribed by the Courant Criterion (Eq. (4)).c=SYNSUB1?y=Kf+y=Kg+DCprime=Ee: (7)Using time line interpolation, the level of SOHt is set withinall the system elements modelled using the highest propagation, resulting in the lowest value of SOHt. Determinationof H and Q is undertaken as illustrated in Fig. 9 for thehorizontal pipework as here the propagation velocity isthat used to set SOHt. As the propagation velocity is lowerin the stack it means that it takes longer than SOHt for pressure changes to be municated to point P from adjacent1Although the equation includes the eVTect of the pipe material, thisis known to be limited when the CRow contains entrained air [9].Fig. 10. Details of the application of the method of characteristics using time line interpolation. (Note: For clarity only alternate nodes arerepresented.)nodes. Depending on the amount of air in the CRow thepropagation and CRow velocities may also bee parable, therefore the approximation represented in Eq. (4)is invalid. These factors mean that if nodes i?1 and i+1are still to be used in the determination of Q and H atthe point in time and space P, the known values of Qand H at these nodes must be obtained more than SOHt before the time plane in which point P exists. This situationmeans that the solution method outlined in Fig. 9 must bemodiFFed, and time line interpolation may be introducedto solve the characteristic equations for Q and H in thesuccessive SOHt time solution planes. Fig. 10 illustrates thetime line interpolation method as applied to this condition.Time line interpolation means that rather than using theprevious time step, and municating the conditions atthat juncture to the current time step, data is conveyedfrom a position m + SI time steps2prior to the currentposition where the characteristic lines (C+and C?) crosspreceding and subsequent nodal planes, respectively.The solution structure is now in place and SIPHONETcan now begin solving for Q and H at each node for eachsuccessive time step. SIPHONET also tracks the movement of the air pocket lodged upstream of the hydraulicjump, as it moves through the system at the ambient CRowvelocity, and the volume is adjusted according to the gaslaw as it moves through the system at puted VSOHtspatial intervals. As the air pocket enters the stack the resultant reduction in the CRow density within the stack generates a partial repressurisation of the system. Th