An efficient numerical method for hydraulic transient computations M. Ciccotelli," S. Sello," P. Molmaro& " CISE Innovative Technology, Segrate, Italy

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1 An efficient numerical method for hydraulic transient computations M. Ciccotelli," S. Sello," P. Molmaro& " CISE Innovative Technology, Segrate, Italy Abstract The aim of this paper is to present a new and more suitable numerical scheme to solve efficiently the Allievi's equations governing the dynamics of fluids inside a pipe. The method arises from a proper combination of the Method Of Characteristics with a spline "shape preserving" interpolation and a nonlinear filtering procedure designed in order to eliminate the residual oscillations. 1 Introduction Problems of fluid-structure interaction or safety monitoring of penstocks need accurate and efficient numerical schemes for hydraulic transient computations. For the solution of these hyperbolic problems, the most common algorithm used is the well known "Method Of Characteristics" (MOC) because of its easy implementation as well as a low time consuming capability. Unfortunately with high gradients and discontinuities the accuracy is affected by significant overshootings. Among the enhancements proposed in literature, we recall "high resolution schemes" based on concepts like "upwinding" and "field by field" decomposition, [1]. These methods, in general, suffer from the disadvantage of a low computational efficiency and notable complexity, especially for practical applications. On the other hand, interpolation of the solution is required using MOC when grid and computational points does not match. Preliminary results, only based on splines "shape preserving" interpolation technique (SSP) have shown a lack of accuracy due to residual overshootings near discontinuities. Such spurious oscillations can be suppressed by means of a spatial nonlinear filtering procedure nested with MOC and SSP

2 388 Hydraulic Engineering Software 2 Numerical methods for the solution of the AllievFs equations The dynamics of a fluid in a straight circular pipe is described in time (t) and space (jc) by its velocity (V) and hydraulic head (H) fields, governed by the Allievi's equations: OH + v + = v dt dx g dx doc (i) g dt g dx dx 2Dg where c, D, z, /, g are the wave celerity, pipe diameter, vertical coordinate, friction coefficient, gravity acceleration respectively. In its simplest version MOC subdivides (usually uniformly) the space domain into intervals of amplitude Ac; the time step At is related to grid spacing Ac through the well known Courant condition: Af < A*Cf n\ W c + V When C,.= l, the positive characteristic lines intersect the time-space grid exactly in computational nodes, where V and H are already known from the previous step. If C,< 1 the need of fields V and H interpolation arises. Indeed this is the case of fluid-structure interaction problems (steel-water) where, for semplicity, spatial grid sizes are the same for both fluid and structural domains; consequently, time step amplitude is forced by the structural dynamics, resulting about 1/4 of thefluidone. For this purpose several schemes have been proposed in literature, leading to some accuracy problems. For example a linear interpolation scheme is a very dissipative one while quadratic interpolation schemes create strong overshootings. A very interesting approach has been proposed by Sibetheros et al. [2]. They proposed a spatial interpolation of V and H by means of spline functions with "shape preserving" capability; these special functions have the peculiarity of preserving the concavity of the data. Such an effect is obtained computing suitable additional grid points whenever very steep slopes in the unknown fields are encountered. A particular feature of this approach is that the Courant condition is no more a stability limit. In fact numerical tests have pointed out that solutions are stable and accurate also for At> Ax/(c+V). In order to check the validity of SSP method several benchmarks have been designed and faced. A comparison of the results has been done with a TVD method showing that the accuracy of SSP method is usually good and in particular that: the amplitude accuracy of SSP depends on the value of the ratio A/7 Ac; in particular is high for low values of the ratio and tends to have a local minimum

3 Hydraulic Engineering Software 389 for ratio values near unity; in this range residual overshooting can be observed, while TVD method is always accurate, the phase accuracy is very good over the whole range of At/Ax (from 0.2 to 1.2), while for TVD method is not, showing a considerable phase shift for low values (0.2). Consequently, some final considerations can be drawn: SSP is an interesting method, with a very good phase behaviour, allows the use of large time steps, shows high frequency residual oscillations. These drawbacks can be overcome by mean of a suitable numerical filtering algorithm. 3 Nonlinear Filtering Approach for the Suppression of Numerical Oscillations The idea of using nonlinear filtering procedures to eliminate or reduce numerically generated oscillations was early investigated by B. Engquist et al. [3] in More recently, Shyy et al. [4] (1992), evaluated the effectiveness of the filter carrying out a detailed study of the development of numerical oscillations generated by different dispersive and diffusive numerical methods, and their interactions with the filter action. Following these previous experiences, we decided to employ a similar nonlinear filtering approach to solve efficiently the Allievi's equations, (1). To check the general effectiveness of the filter, we analysed the application of several extended versions of the original scheme, such as TVD limiters or relaxation procedures, [4] Fundamentals The purpose of our approach is the coupling of a generic numerical integration scheme G( -) with a given filter operator P( ), [3]: where: v"+* represents the first approximated solution and u^^ the filtered one. For a filter P to be effective it should possess some basic properties to provide a minimal effect on an already smooth solution and to achieve an efficient action, with a minimal computational effort, to guarantee no spurious oscillations near discontinuities, [3]. Let us consider briefly the fundamental steps in the nonlinear filtering procedure, starting from the simpler version, with no TVD constraint. Given the vector solution: \ = (v^\ v^,..., v,/"^)\ from the first of eqns (3), we start with the following procedure: 1) search for the spatial components of the vector v, v/^\ j = l,...,n, with corrections of local extrema: (A+Vj"~^)(A-Vj"+*)<0, with: A+, A., the usual forward and backward differences respectively: A±VJ=±(VJ+I-VJ),

4 390 Hydraulic Engineering Software 2) the value corrections in step 1) are made by decreasing maxima and increasing minima, 3) when a correction is added to a grid pointy*, the same correction must be subtracted from a neighboring point in such a way to preserve a conservation criterion. More precisely, the corrected neighbor is the grid point j + 7*, or j- 7*, if the absolute maximum of differences: A+ and A coincides with: A+, or A., respectively, 4) no value may be corrected to overcome its neighbors. The filtering scheme at a given time step may be summarized as follow: 6 = min(6_, 6+/2) v. = v. + s 5 where the corrections are performed inside an iterative cycle in the spatial domain, and: 6* / I A W + l I I * W + l I _ = Engquist et al. showed that the above no TVD filtering scheme does not affects the convergence property of the original numerical scheme. The nonlinear filtering procedure has proved to be very effective in the suppression of spurious, numerically generated, oscillations of 2 Ax and 4Ax wavelengths, [4]. As the wavelengths become longer the filter effectiveness become smaller. The inability of the filter to treat long wavelength oscillations may results a pleasing feauture, because generally we aim to eliminate short wavelengths oscillations produced by the numerical scheme while only slightly to affect physically expected longer wavelength components. Regarding the procedure of practical filter implementation, since the initially generated oscillations tend to develop with time, the filtering action is most effective when applied at each time step of the computation. 3.2 Filtering improvements Many extensions of the above filter are in principle possible. In fact, the filtering procedure does not yield unique solutions with different degrees of dissipation and dispersion characteristics of a given numerical scheme. A relaxation procedure, for example, has been proposed by Shyy et al. [4] in order to enhance the effectiveness of the filtering. More precisely that improvement consists in the introduction, at the correction level, of a multiplicative parameter, w (relaxation parameter), mostly adjusted empirically. In [4] the authors showed that a w> 1, further improves the filtered solution. Indeed, the relaxation procedure introduces extra adjustments by removing the plateaus, i.e. the multipoints extrema:

5 Hydraulic Engineering Software = min(6_, 5J2) = v. + a) s 6 with: w>0. Another possible extension of the filtering consists in the generalization of the reduction coefficient for the quantity 5+, with numerical values different than 0.5. That version further enhances the effectiveness of the procedure, especially when residual plateaus are present: 6=min(<5_,a<5+) with w>0 and In order to eliminate some unwelcome properties of the basic filter, in the present work we consider also an important modification, first suggested by Engquist [3]. The aim is to modify the algorithm to obtain a TVD-enforcing filtering. Now the extremum condition at grid point JQ is replaced by the following one: min(v,\, va v,\) < v."""\ if. v"** is a local minimum, JQ * JQ JQ JQ JQ 6 max(vy"_i, Vy", v"+i) > v"*, if. v"* is a local maximum, for all extrema: 1^""^ of v"+\ Moreover for a TVD constraint a condition of no consecutive extrema is also necessary; that implies the replacement of the expression: 6=min(6_,a<5+) with the following one: where: 6 = min(6_, && i M+l, n n n \ d^ = Vj - max(v,-_i, Vj, v^ n n ^ n+l _i, v,, ^+J - v^ The final result is an accurate nonlinear filtering procedure with TVD condition. As already pointed out in [3,4], the above more sophisticated versions of the filtering procedure appear less robust than the first proposed one. Indeed at present does not exist general theorems proving that the TVD version is a stable finite algorithm, i.e. the filter terminates in a finite number of iterations on each time level, even if there are very small changes in the algorithm structure. Therefore it seems very opportune to carry out extensive theoretical and numerical analyses to clarify and to enlighten some important properties of the above appealing extensions of nonlinear filters.

6 392 Hydraulic Engineering Software 4 Applications Our test problem consists in a straight pipe (D=0.4 [m], lenght L=100 [m]) conveying water flux at V^ =.5 [m/s] from a reservoir to a valve suddenly closed at the beginning of the transient. Other parameters are: c=1450 [m/s], /=0.5, Ax=L/20 [m], C,=0.9. A parametric analysis has been carried out using different combinations of TVD and relaxation parameter, as well as different grid spacings. On the other hand only a representative result will be shown in the following. In Fig. 1 typical pressure time behaviours, obtained with SSP and SSPF (with relaxation, o> = 1.2, and TVD constraint), are reported. As anticipated SSP exhibits a strong residual overshootings near discontinuities, expecially at the beginning of the transient. Instead SSPF almost eliminates all the spurious oscillations without affecting the phase properties of the solution. Fig. 2 shows an enlargement of the transient start of Fig. 1: it is evident how residual oscillations are almost completely suppressed. Moreover the capability to make very sharp bends near all discontinuities is clearly pointed out. Filter action can be better enlighted through time evolution of the wave spatial profiles. In fact, Fig. 3 shows how the spurious oscillations arise just the transient starts and how they tend to amplify. On the other hand, in Fig. 4 spurious oscillations are completely absent. It is worth noting that this notable result has been reached applying the filtering technique each time step. 5 Conclusions The principal features of the proposed method make it suitable for all the wave propagation-like problems where a high level of accuracy and efficiency is required. The new scheme has been checked facing some significant test problems, mainly hydraulic shock wave propagation ones, obtaining efficiently, very accurate results. Some applications of the numerical technique to real systems, related to fluid-structure interaction and penstock monitoring cases, are in progress. References 1. Rider, W.J. Methods for extending high-resolution schemes to nonlinear systems of hyperbolic conservation laws, Int. Jour. Num. Meth. Fluids, 1993, Sibetheros, I.A., Hooley, E.R. and Branski, J.M. Spline interpolations for water hammer analysis Journal of Hydraulic Engineering, Vol. 117, No 10, Engquist, B.,L6tstedt, P. and Sjogreen, B. Nonlinear filters for efficient shock computation, Mathematics of Computation, 1989, 52, Shyy, W., Chen, M.H., Mittal, R. and Udaykumar, H.S. On the suppression of numerical oscillations using a nonlinear filter, Jour. Comp. Phys., 1992, 102.

7 Hydraulic Engineering Software 393 SRSF MetH->o<d ~WD o r-ne<ga = 1.2 Figure 1: Pressure time evolution at closed end computed by SSP (dashed line) and SSPF (continuous line). Pressure is expressed in Pa (N/nf) and time in sec. PSF Metnoa TVD ome<ga=1.2 Figure 2: Particular of Fig. 1: SSP (dashed line) vs. SSPF (continuous line).

8 394 Hydraulic Engineering Software SRS Method Space Profile Figure 3: Pressure wave spatial profiles at different time steps with SSP. x 1 O SRSF Method (TVO) Space Spaco Figure 4: Pressure wave spatial profiles at different time steps with SSPF (TVD)