1,721,056 research outputs found
A new well-balanced Hermite weighted essentially non-oscillatory scheme for shallow water equations
No full textHermite weighted essentially non-oscillatory (HWENO) methods were introduced in the literature, in the context of Euler equations for gas dynamics, to obtain high-order accuracy schemes characterized by high compactness (e.g. Qiu and Shu, J. Comput. Phys. 2003; 193:115). For example, classical fifth-order weighted essentially non-oscillatory (WENO) reconstructions are based on a five-cell stencil whereas the corresponding HWENO reconstructions are based on a narrower three-cell stencil. The compactness of the schemes allows easier treatment of the boundary conditions and of the internal interfaces. To obtain this compactness in HWENO schemes both the conservative variables and their first derivatives are evolved in time, whereas in the original WENO schemes only the conservative variables are evolved.
In this work, an HWENO method is applied for the first time to the shallow water equations (SWEs), including the source term due to the bottom slope, to obtain a fourth-order accurate well-balanced compact scheme. Time integration is performed by a strong stability preserving the Runge–Kutta method, which is a five-step and fourth-order accurate method. Besides the classical SWE, the non-homogeneous equations describing the time and space evolution of the conservative variable derivatives are considered here. An original, well-balanced treatment of the source term involved in such equations is developed and tested. Several standard one-dimensional test cases are used to verify the high-order accuracy, the C-property and the good resolution properties of the model
SCHEMI WENO BILANCIATI DEL QUARTO ORDINE PER L’IDRODINAMICA ALLE ACQUE BASSE
No full textNel presente lavoro vengono descritti due modelli, appartenenti alla classe degli schemi WENO, accurati al quarto ordine nello spazio e nel tempo, per l’integrazione numerica delle equazioni alle acque basse comprensive del termine sorgente dovuto alla pendenza del fondo. In entrambi i modelli è di nuova concezione il trattamento del termine sorgente che soddisfa la C-property, la proprietà di preservare lo stato di quiete, e contemporaneamente mantiene l’elevato ordine di accuratezza. L’introduzione di tale trattamento permette l’applicazione degli schemi WENO a problemi caratterizzati da altimetrie del fondo estremamente irregolari. Nel primo modello, di tipo centrato e basato sull’utilizzo di griglie sfalsate, l’accuratezza temporale è ottenuta ricorrendo ad uno schema Runge-Kutta (RK) accoppiato con l’approccio Natural Continuous Extension (NCE). L’accuratezza spaziale è ottenuta utilizzando opportune ricostruzioni WENO delle variabili conservative e del carico piezometrico. Il trattamento originale del termine sorgente coinvolge due procedure. La prima riguarda la valutazione congiunta delle derivate puntuali del flusso accoppiate con il valore puntuale del termine sorgente. La seconda coinvolge l’integrazione spaziale del termine sorgente, manipolato analiticamente al fine di trarre vantaggio dalla maggiore regolarità della quota della superficie libera rispetto alla quota del fondo. Nel secondo modello, di tipo upwind, il quarto ordine di accuratezza temporale è ottenuto ricorrendo ad uno schema Runge-Kutta Strong Stability Preserving, RKSSP(5,4). L’accuratezza spaziale è ancora ottenuta utilizzando ricostruzioni WENO delle variabili conservative e della quota della superficie libera. Il trattamento del termine sorgente, ancora bilanciato e di alto ordine, coinvolge la sola integrazione spaziale del termine sorgente ed è compiuta ricorrendo ad una procedura analoga a quella del primo modello. Diversi casi test classici sono stati utilizzati per verificare l’ordine di accuratezza dei modelli, l’esatto soddisfacimento della C-property e la buona risoluzione delle discontinuità
Hydraulic jump in diverging channels
No full textThe classical problem of the hydraulic jump in diverging channels is revisited and reformulated, in terms of linear and angular momentum conservation, at the integral scale. The former balance is used in its well known form, while the latter is expressed in the simplest meaningful formulation, taking into account of the moments exerted by momentum flux, by pressure force in each cross section and on lateral walls, and by vertical stresses. The whole flow is supposed to be ideally divided in a mainstream, conveying the total discharge, and in a roller, exerting stresses on the mainstream.
The scheme is the axially-symmetric counterpart of the 2D plane scheme formulated by Valiani [Linear and angular momentum conservation in hydraulic jump, JHR, 35(3), pp.323-352, 1997]. Gravity forces are supposed to be exactly counteracted by bottom reactions. Outside the jump, the inviscid solution is assumed to minimize the degrees of freedom of the problem. The obtained system of two conservation laws, together with the appropriate boundary conditions, is not analytically solvable due to the highly nonlinear relationship between the vertical length scale (depth) and the longitudinal length scale (radius), but a simple numerical solution gives the sequent depths and their positions, as functions of the non-dimensional discharge and of the downstream/upstream energy ratio of the flow. Taking into account the uncertainties in defining the downstream cross section of the jump, the comparison between the numerical solution and a selected set of laboratory data shows the reliability and accuracy of the proposed scheme
A well-balanced, third-order-accurate RKDG scheme for SWE on curved boundary domains
No full textIn this work, we present a third-order-accurate Runge-Kutta discontinuous Galerkin model defined on an unstructured triangular grid. The originality of this model lies in its ability to correctly and efficiently treat computational domains with curved boundaries while preserving the exact well-balancing between the source terms and the flux divergence in the case of quiescent flow. With this aim, the model uses straight-sided elements in the inner part of the domain and curved-sided elements in the area adjacent to the boundaries, limiting as much as possible the additional computational costs due to the use of high-order isoparametric elements. Together with a careful combination of well-known techniques used for the exact balancing of the scheme with straight-sided elements, an original solution is proposed to achieve the same balancing for the curved-sided elements. Such an approach is based on a locally modified formulation of the shallow water equations. Proofs of the consistency between the modified formulation and the original one, of the satisfaction of the C-property for quiescent flow and of the negligibility of the added new terms in the case of moving water are provided.
Several examples are presented to demonstrate the well-balancing property and the overall good behavior of the proposed scheme
Modellazione numerica bidimensionale alle acque basse del deflusso in presenza di ostacoli isolati o aggregati
No full textIl lavoro consiste nell’applicazione di un codice di calcolo autoprodotto ai volumi finiti, idoneo ad integrare numericamente le equazioni bidimensionali alle acque basse, a casi in cui un’onda di sommersione investa ostacoli isolati o aggregati che modifichino significativamente il campo di moto. Il codice di calcolo è del tipo di Godunov, esplicito, ai volumi finiti, accurato al secondo ordine nello spazio e nel tempo. Il solutore approssimato di Riemann utilizzato è quello di Harten, Lax e van Leer. Vengono studiati casi tipo nei quali un’onda a fronte ripido investe ostacoli isolati a pianta quadrata, con lati paralleli o disposti a 45° rispetto al fronte incidente. Si evidenziano la diffrazione provocata dagli ostacoli, il ruolo delle pareti laterali nella riflessione delle ondulazioni, le principali caratteristiche delle ondulazioni stesse. Vengono inoltre simulate alcune tipi-che onde di sommersione che investono gruppi di ostacoli che possano schematicamente rappresentare aggregazioni di edifici. I risultati delle simulazioni vengono confrontati con risultati sperimentali di laboratorio ottenuti presso l’Enel.Hydro di Milano
Un modello discontinuous Galerkin per la simulazione del deflusso in tubi collassabili con proprietà meccaniche discontinue: applicazione allo studio dell’emodinamica
No full textIn questo lavoro viene presentato un nuovo modello numerico monodimensionale Runge-Kutta discontinuous Galerkin, accurato al terzo ordine nello spazio e nel tempo, per l’integrazione delle equazioni del moto di un fluido incomprimibile in una condotta in pressione. Il problema è reso complicato dall’ipotesi che il tubo possa essere collassabile ed abbia caratteristiche meccaniche e geometriche variabili nello spazio, anche in modo discontinuo. La naturale applicazione del modello è la simulazione del deflusso sanguigno nel sistema circolatorio, con particolare enfasi al moto del sangue nelle vene (Fung, 1997) e in vasi riparati chirurgicamente tramite l’impianto di protesi vascolari (e.g. Umscheid & Stelter, 1999)
Linear and angular momentum conservation for the hydraulic jump in converging channels
No full textThis note is the final completion of a previously published work concerning the integral conservation of linear and angular momentum in the steady hydraulic jump in a linearly diverging channel [Valiani, A., Caleffi, V. (2011). Linear and angular momentum conservation in hydraulic jump in diverging channels. Adv. Water Res. 34(2), 227–242]. The same reasoning is applied to a linearly converging channel, and the theoretical framework, which is almost completely the same, is shown to remain valid. Using a proper mechanical scheme, an analytical solution is obtained for the free surface profile of the flow. This solution allows the determination of the sequent depths and their
positions. Thus, the length of the jump, which is assumed to be equal to the length of the roller, is also found. The mainstream and roller thicknesses can also be derived. This model may be used to derive the average shear stress exerted by the roller on the mainstream and the related exact expression for the total power loss in the jump, allowing to demonstrate the internal consistency of the proposed conceptual scheme
A discontinuous Galerkin scheme for the simulation of flows in collapsible tubes with discontinuous mechanical properties
No full textIn this work, we present a one-dimensional numerical model for the hyperbolic balance laws related to the incompressible fluid flow in a pipe.
The problem is made challenging by the hypothesis that the tube can be collapsible and characterized by variable mechanical and geometrical properties. In order to increase the generality of the model, we assume that the variation of the properties of the tube can be discontinuous [1]. Moreover, the nature of the adopted tube-law, describing the relation between the pressure and the cross-section, leads to the presence of non-conservative products in the balance laws.
The model belongs to the family of the Runge-Kutta discontinuous Galerkin schemes [2] and is third order accurate in space and time. To cope with the presence of the non-conservative products, a path-consistent approach is followed [3] and the DOT Riemann solver is here applied [4]. The use of a linear path is not sufficient to achieve satisfactory results; therefore, the curvilinear path proposed by Müller & Toro [5] is selected. To avoid unphysical oscillations, a WENO limiter is included in the model. Interestingly, the application of standard WENO techniques are not suitable for this particular scheme and a modified approach is here proposed.
The natural application of the model is the simulation of blood flow in the circulatory system, with particular emphasis on the blood flow in the veins and in vessels surgically repaired by implantation of vascular prostheses.
References
1. E. F. Toro and A. Siviglia. Flow in Collapsible Tubes with Discontinuous Mechanical Properties: Mathematical Model and Exact Solutions. Communications in Computational Physics, 13(2):361–385, 2013.
2. B. Cockburn and C. W. Shu. Runge-Kutta Discontinuous Galerkin Methods for Convection-dominated Problems. Journal of Scientific Computing, 16(3):173–261, 2001.
3. M. Parés. Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework. SIAM Journal on Numerical Analysis, 44:300–321, 2006.
4. M. Dumbser and E. F. Toro. A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems. Journal of Scientific Computing, 48(1-3):70–88, 2011.
5. L. O. Müller and E. F. Toro. Well-balanced high-order solver for blood flow in networks of vessels with variable properties. International Journal for Numerical Methods in Biomedical Engineering, 29:1388–1411, 2013
A RKDG scheme for 2D SWE on curved boundary domains
No full textThis work regards a high-order numerical scheme for the integration of the two-dimensional Shallow Water Equations (SWE). First, we highlight the importance of the high-order representation of the boundaries when the numerical scheme has a high-order accuracy.
We show the impossibility to obtain a physically realistic solution for some families of steady-state problems using only straight-sided elements. Moreover, we prove that the existing techniques are not suitable to obtain a well-balanced model for curved boundaries.
Our solution to the examined problems consists of a third-order-accurate, well-balanced, Runge-Kutta discontinuous Galerkin (RKDG) model for the integration of the SWE on an unstructured triangular grid. The problem of the curved boundary is addressed by using a proper mixture of straight-sided elements, in the inner part of the computational domain, and of elements with a single curved edge, in the regions near the boundaries.
A careful combination of available techniques yields a well-balanced model on the straight-sided elements, and an original approach is proposed for balancing the model on the curved-sided elements. This approach is based on a modified mathematical model of SWE which is consistent with the original one and allows to achieve the well-balancing property
A 2D local discontinuous Galerkin method for contaminant transport in channel bends
No full textIn this work, a third-order local discontinuous Galerkin (LDG) method is applied to the numerical integration of a two-dimensional, depth-integrated mathematical model escribing the flow hydrodynamics and the transport of a passive contaminant in open-channel bends.
The mathematical model is described in Begnudelli et al. (2010) [14] and treats the main physical aspects of the flow in curved channels (including bottom shear, momentum dispersion, scalar dispersion and turbulent diffusion) with a homogeneous degree of complexity and exhaustiveness. The numerical integration is performed using the scheme presented in Caleffi and Valiani (2012) [27], which is extended in this work to allow the treatment of diffusion terms. The capability of the model to correctly and efficiently treat computational domains with curved boundaries while preserving the exact well-balancing property is of fundamental importance for the correct simulation of the free-surface flow in laboratory flumes and real channels.
Three test cases are used to validate the model. The first case consists of a problem, with an analytical solution, related to the tracer convection–diffusion in a uniform flow. This test is selected to highlight the excellent resolution obtainable using an LDG method. The other test cases, consisting of comparisons
between numerical results and laboratory data, are selected to verify the capability of the model to reproduce real world phenomena. Both steady and unsteady tracer dispersion are taken into account.
The results show the potentiality of the high-order accuracy models when applied to engineering problems that are characterized by inherently complex physical phenomena. The results also confirm the possibility of using extremely coarse grids, which leads to a high computational efficiency, in these
real-world applications. Moreover, the good agreement between the experiments and simulations is a further confirmation of the suitability of two-dimensional depth-averaged models for the study of the convection–dispersion of momentum and pollutants in curved channels and rivers.
Finally, the reconstruction of the solution by polynomial shape functions, which are typical of the LDG schemes, allows the streamline curvature, which is necessary to evaluate the diffusion coefficients, to be computed in a self-consistent manner, without the use of arbitrary reconstructions
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