TVD-type Scheme Research Articles

A 2DH fully dispersive and weakly nonlinear Boussinesq-type model based on a finite-volume and finite-difference TVD-type scheme

In this study, we developed a fully dispersive Boussinesq-type wave model in the modelling framework of the public-domain Boussinesq model, FUNWAVE-TVD. The model adopts the fully dispersive Boussinesq equations of Karambas and Memos (2009) and uses a finite-volume and finite-difference TVD-type scheme. A well-balanced conservative form of the governing equations is derived to facilitate the hybrid numerical scheme. Flux terms were computed by the MUSCL-TVD scheme up to the fourth-order accuracy within the Riemann solver. The third-order Strong Stability-Preserving (SSP) Runge–Kutta scheme was used for time stepping. The convolution integral terms were estimated by the numerical evaluation and the spatial derivative terms were computed by the finite difference scheme. Wave breaking is predicted by locally switching the Boussinesq equations to nonlinear shallow water equations with a Froude number criterion. The model is validated against the linear wave theory and various experiments to examine the capability of the model in simulating wave dispersion, shoaling, breaking, refraction, diffraction, and run-up.

NearCoM-TVD — A quasi-3D nearshore circulation and sediment transport model

The newly developed nearshore circulation model, SHORECIRC, using a hybrid finite-difference finite-volume TVD-type scheme, is coupled with the wave model SWAN in the Nearshore Community Model (NearCoM) system. The new modeling system is named NearCoM-TVD and the purpose of this study is to report the capability and limitation of NearCoM-TVD for several coastal applications. For tidal inlet applications, the model is verified with the semi-analytical solution of Keulegan (1967) for an idealized inlet-bay system. To further evaluate the model performance in predicting nearshore circulation under intense wave–current interaction over complex bathymetry, modeled circulation patterns are validated with measured data during RCEX field experiment (MacMahan et al., 2010). For sediment transport applications, two sediment transport models are applied to predict three sandbar migration events at Duck, NC, during August to October 1994 (Gallagher et al., 1998). The model of Kobayashi et al. (2008) incorporates wave-induced onshore sediment transport rate as a function of the standard deviation of wave-induced horizontal velocities. The modeled beach profile evolution for two offshore events and one onshore event agrees well with the measured data. The second model investigated here combines two published sediment transport models, namely, the total load model driven by currents under the effect of wave stirring (Soulsby, 1997) and the wave-driven sediment transport model due to wave asymmetry/skewness (van Rijn et al., 2011). The model study with limited field data suggests that the parameterization of wave stirring is appropriate during energetic wave conditions. However, during low energy wave conditions, the effect of wave stirring needs to be re-calibrated.

Simulation of Viscous and Reactive Hypersonic Flows Behaviour in a Shock Tube Facility: TVD Schemes and Flux Limiters Application

Work performed in this study concerns mainly the analysis and the wisely use of TVD type schemes (total variation diminishing) for numerical simulation of reactive flows, these schemes are first presented in scalar equation. Their extension to Euler equations for a reactive gas mixture is conducted through the approximate extended solver of Riemann problem. A comparative study of specific variants of TVD schemes has been made in the case of one-dimensional unsteady flow for an inert and reactive gas mixture, which represents the classical instance of a shock tube. The purpose of this investigation is to highlight the general behaviour (order of accuracy) and performance of TVD schemes with various flux limiters for the simulation of reactive flows and in particular, to make possible the capture of the shock wave together with waves expansion for choosing the appropriate scheme to apply eventually in simulation of hypersonic viscous flow in chemical non equilibrium.

A high-order adaptive time-stepping TVD solver for Boussinesq modeling of breaking waves and coastal inundation

We present a high-order adaptive time-stepping TVD solver for the fully nonlinear Boussinesq model of Chen (2006), extended to include moving reference level as in Kennedy et al. (2001). The equations are reorganized in order to facilitate high-order Runge–Kutta time-stepping and a TVD type scheme with a Riemann solver. Wave breaking is modeled by locally switching to the nonlinear shallow water equations when the Froude number exceeds a certain threshold. The moving shoreline boundary condition is implemented using the wetting–drying algorithm with the adjusted wave speed of the Riemann solver. The code is parallelized using the Message Passing Interface (MPI) with non-blocking communication. Model validations show good performance in modeling wave shoaling, breaking, wave runup and wave-averaged nearshore circulation.

Applying the TVD scheme to calculate two-phase flows with different velocities and pressures of the components

The TVD-type difference scheme and the large particle method are studied and realized for solving the mathematical models of mechanics of heterogeneous media with different velocities and pressures of the components. The results calculated in accordance with the TVD scheme and the modified large-particle method are compared with exact solutions for two types of shock-wave configurations in heterogeneous media. It is shown that for this class of the problems of mechanics of heterogeneous media the TVD-type schemes are more effective with respect to such criteria as the computation time and the grid step. By way of illustration, the problem of shock wave reflection from a wall in a concentration-inhomogeneous medium is considered.

Turbulent transonic flow simulation using a pressure-based method

In the past, turbulent compressible aerodynamic flow calculations were usually carried out with density-based numerical methods and zero-equation turbulence models. However, pressure-based numerical methods and more advanced turbulence models have been routinely used in industry for many internal flow simulations and for incompressible flow calculations. The application of pressure-based numerical method and advanced turbulence models to the calculations of aerodynamic flows is still not well demonstrated and accepted. In this study, an advanced pressure-based numerical method and a recently proposed near-wall compressible two-equation turbulence model are used to calculate external transonic aerodynamic flows. Several TVD-type schemes are incorporated into the pressure-based method to better capture discontinuities such as shocks. A compressible near-wall two-equation turbulence model is implemented into the numerical scheme to calculate transonic turbulent flows over NACA 0012 and RAI2822 airfoils with and without shocks. Good correlations are obtained with wind tunnel data as well as with results calculated using the Baldwin-Lomax model. Therefore, this study demonstrates that advanced turbulence models capable of handling flows with separation and recirculation could be implemented with ease to a pressure-based method with improved capability to resolve discontinuities to calculate turbulent compressible aerodynamic flows.

A multigrid Tvd-type scheme for computing inviscid and viscous flows

A numerical scheme for the computation of steady-state transonic flow fields is presented, which is based on a total variation diminishing (TVD) approach. Various kinds of the anti-diffusive flux terms have been considered, and their effect on the computed results investigated. The time-dependent governing equations are given in a conservative formulation and solved by a hybrid multistage Runge-Kutta scheme. To obtain an improved convergence rate a multigrid procedure has been added to the scheme. The time-marching method presented has been verified by inviscid and viscous two-dimensional flow-field computations. It has been found that a simple laminar shock boundary-layer interaction problem serves as a good indicator for the effect of the various flux limiters of a TVD-scheme. On the whole, it has been observed that for the solution of the Euler equations the most diffusive of the limiter functions is capable of yielding a good shock resolution. However, the flux limiters have to be chosen very carefully if viscous flow with separation bubbles is to be considered. The better the shock resolution of the flux limiter applied, the greater the effect the various time-stepping procedures have on the results.

Underexpanded Impinging Jet. Numerical Solution of Eulder Equations.

The underexpanded jet which is exhausted from the circular nozzle is studied numerically in this investigation. The jet impinges on a flat plate located normal to the jet axis. The axisymmetric Euler equations are solved using a multigrid TVD-type scheme. Jets with different pressure ratios and nozzle-plate distances are simulated. Comparing the numerical results with the experimental ones, good agreement is found and, furthermore, the effect of the nozzle-plate spacing on the wave pattern in the jet can be described. The formation of the separation bubble at a particular nozzle-plate distance is also discussed.