A Multidimensional Flux Function with Applications to the Euler and Navier-Stokes Equations

doi:10.1006/jcph.1993.1077

Journal of Computational Physics

Volume 105, Issue 2, April 1993, Pages 306-323

https://doi.org/10.1006/jcph.1993.1077 Get rights and content

Abstract

A grid-independent approximate Riemann solver has been developed for use in both two- and three-dimensional flows governed by the Euler or Navier-Stokes equations. Fluxes on grid faces are obtained via wave decomposition. By assuming that information propagates in the velocity-difference directions, rather than in the grids-normal directions as in a standard grid-aligned solver, this flux function more appropriately interprets and hence more accurately resolves shock and shear waves when these lie oblique to the grid. The model, which describes the difference in states at each grid interface by the action of five waves, produces a significant increase in accuracy for both supersonic and subsonic first-order spatially accurate computations. Second-order computations with the grid-independent flux function are generally only worth the additional effort for flowfields whose primary structures include shear waves that lie oblique to the grid. Included in this category is the viscous flow over an airfoil in which a separated shear layer emanates from the airfoil surface at an angle to the grid. Pressure distortions which can result from misinterpretation of the oblique waves by a grid-aligned solver are essentially eliminated by the grid-independent flux function.

References (0)

Cited by (49)

A study of higher-order reconstruction methods for genuinely two-dimensional Riemann solver
2021, Journal of Computational Physics
Citation Excerpt :
Early successes emerged with the work of Collela [12], LeVeque [13] by building some level of multidimensionality out of the one-dimensional Riemann solvers. More attempts include the introductions of the multidimensionality into the one-dimensional Riemann solvers by Roe and Rumsey [14,15], the multidimensional HLLE scheme proposed by Wendroff [16], and the multidimensional extensions of Roe's linearized Riemann solvers by Abgrall [17,18], Fey [19,20], and Brio [21]. However, these schemes have been restricted for application because of their complex forms and high computational cost [22].
Balsara's genuinely multidimensional Riemann solvers, which possess a simple closed-form and high efficiency in the low-order cases, realize their multidimensional effect successfully by introducing the vertex-based framework. However, traditional higher-order (third-order and above) reconstruction methods built for the structured grids become ambiguous when applied directly into these solvers. Nowadays, there is little research on the higher-order reconstruction methods for such Riemann solvers besides Balsara's work by adopting the ADER-WENO schemes. Based on Balsara's work, we conduct research on the multidimensional higher-order reconstruction procedure in this study. This paper investigates two third-order reconstruction methods with the WENO approach for the genuinely two-dimensional HLLE Riemann solver. The critical idea of the reconstruction methods is to utilize the solution or its derivative to construct the spatial two-dimensional interpolation polynomials with the third-order accuracy. The first reconstruction method denoted by TWENO constructs the polynomials by the Taylor expansion, while the second one denoted by HWENO adopts the Hermite interpolation polynomials. By combining the polynomials with the WENO limiter, these schemes can provide required reconstructed values at the midpoints and corner-points of the cell interfaces for the framework and avoid oscillations near discontinuities. A discontinuity-detector technology is adopted to improve the computational efficiency of these reconstruction methods in simulating complex flows. Several numerical test cases are conducted and show that these third-order methods achieve their design accuracy and capture the shock waves with no overshoots or oscillations. The multidimensional nature of the solutions is kept well by the TWENO/HWENO methods, and the computational time is reduced by about 30% with the discontinuity-detector technology. Moreover, it is found that the genuinely two-dimensional Riemann solver fits the third-order reconstruction methods naturally due to its unique application of a Simpson rule.
Improvement of the genuinely multidimensional ME-AUSMPW scheme for subsonic flows
2021, Computers and Mathematics with Applications
Citation Excerpt :
In order to dispel such problem, many researchers conduct studies on the genuinely multidimensional Riemann solver. For example, Roe and Rumsey reconstructed the Roe scheme from one dimensional problem to multidimensional problem [16,17]. LeVeque proposed the multidimensional wave propagation scheme and Collela developed the corner transport upwind scheme [18,19].
Since proposed, the genuinely multidimensional Riemann solvers have been attracting more attentions because it considers the waves normal and parallel to the cell interfaces at the same time. However, all these methods are built upon the hyperbolic property of the compressible Euler equations. Therefore, they can not be applied to solve the Euler equations in steady subsonic cases which are with the elliptic property. In this study, a novel multidimensional Riemann solver called MULE-AUSMPWM (Multidimensional E-AUSMPW Modified) is proposed. It is built upon the Zha–Bilgen splitting procedure and adopts the Balsara's multidimensional wave model as the ME-AUSMPW scheme. In addition, it improves the ME-AUSMPW scheme's accuracy at subsonic speeds by avoiding simulating the convective fluxes in fully upwind manners. Extensive numerical tests, such as the two dimensional Euler problem, the one dimensional moving contact discontinuity case, the two-dimensional double Mach reflection case, the two-dimensional Riemann problem, and the turbulent flow past a 2d NACA0012 airfoil case, are conducted. Results illustrate that the MULE-AUSMPWM scheme proposed is capable of simulating compressible complex flows and improving the existing multidimensional Riemann solvers' accuracy at subsonic speeds remarkably.
Development of accurate and robust genuinely two-dimensional HLL-type Riemann solver for compressible flows
2020, Computers and Fluids
When using the conventional direction splitting method to calculate multidimensional high speed gasdynamical flows, Riemann solvers capable of resolving contact surface and shear wave accurately will suffer from different forms of shock instability, such as the notorious carbuncle phenomenon. The stability analysis shows that the lack of dissipation in the direction transverse to the shock front leads to the shock instability of low-dissipation HLLEM solver. To overcome this defect, an accurate and carbuncle-free genuinely two-dimensional HLL-type Riemann solver is proposed. Using Zha-Bilgen splitting method, the flux vector of two-dimensional Euler equations is split into the convective flux and pressure flux. An algorithm similar to AUSM+ scheme is adopted to calculate the convective flux and the pressure flux is calculated by the low-dissipation HLLEM scheme. Following Balsara’s idea, the genuinely two-dimensional properties of the new solver are achieved by solving the two-dimensional Riemann problem that considers transversal features of the flow at each vertex of the cell interface. Numerical results of benchmark tests demonstrate that the new solver has higher resolution and better robustness than the conventional HLLEM solver implemented in dimension by dimension.
A new genuinely two-dimensional Riemann solver for multidimensional Euler and Navier–Stokes Equations
2019, Computer Physics Communications
Citation Excerpt :
By considering not only the information propagating normal to the cell interfaces, but also those propagating transverse to the cell interfaces, the multidimensional Riemann solvers are capable of capturing multidimensional complex flows accurately in theory. Following this idea, Roe and Rumsey introduced the multidimensionality into the one-dimensional Riemann solvers [23,24]. Collela proposed the corner transport upwind scheme [25] and LeVeque developed the multidimensional wave propagation scheme [26].
A novel two-dimensional flux splitting Riemann solver called ME-AUSMPW (Multi-dimension E-AUSMPW) is proposed. By borrowing the Balsara’s idea, it is built upon the Zha–Bilgen splitting procedure and considers both the waves orthogonal to the cell interfaces and the waves transverse to the cell interfaces. Systematic numerical cases, including the one dimensional Sod shock tube case, the double Mach reflection of a strong shock case, the two-dimensional Riemann problem, the hypersonic viscous flows over the two-dimensional Double-ellipsoid case, and the shock wave/laminar boundary layer interaction problem, are carried out. Results show that the ME-AUSMPW scheme proposed in this manuscript is with a higher resolution than conventional one-dimension Riemann solvers in simulating not only inviscid complex flows, but also viscous flows. Therefore, it is promising to be widely used in both scholar and engineering areas.
A genuinely two-dimensional Riemann solver for compressible flows in curvilinear coordinates
2019, Journal of Computational Physics
A genuinely two-dimension Riemann solver for compressible flows in curvilinear coordinates is proposed. Following Balsara's idea, this two-dimension solver considers not only the waves orthogonal to the cell interfaces, but also those transverse to the cell interfaces. By adopting the Toro–Vasquez splitting procedure, this solver constructs the two-dimensional convective flux and the two-dimensional pressure flux separately. Systematic numerical test cases are conducted. One dimensional Sod shock tube case and moving contact discontinuity case indicate that such two-dimensional solver is capable of capturing one-dimensional shocks, contact discontinuities, and expansion waves accurately. Two-dimensional double Mach reflection of a strong shock case shows that this scheme is with a high resolution in Cartesian coordinates. Also, it is robust against the unphysical shock anomaly phenomenon. Hypersonic viscous flows over the blunt cone and the two-dimensional Double-ellipsoid cases show that the two-dimensional solver proposed in this manuscript is with a high resolution in curvilinear coordinates. It is promising to be widely used in engineering areas to simulate compressible flows.
A two-dimensional Riemann solver with self-similar sub-structure - Alternative formulation based on least squares projection
2016, Journal of Computational Physics
Just as the quality of a one-dimensional approximate Riemann solver is improved by the inclusion of internal sub-structure, the quality of a multidimensional Riemann solver is also similarly improved. Such multidimensional Riemann problems arise when multiple states come together at the vertex of a mesh. The interaction of the resulting one-dimensional Riemann problems gives rise to a strongly-interacting state. We wish to endow this strongly-interacting state with physically-motivated sub-structure. The self-similar formulation of Balsara [16] proves especially useful for this purpose. While that work is based on a Galerkin projection, in this paper we present an analogous self-similar formulation that is based on a different interpretation. In the present formulation, we interpret the shock jumps at the boundary of the strongly-interacting state quite literally. The enforcement of the shock jump conditions is done with a least squares projection (Vides, Nkonga and Audit [67]). With that interpretation, we again show that the multidimensional Riemann solver can be endowed with sub-structure. However, we find that the most efficient implementation arises when we use a flux vector splitting and a least squares projection. An alternative formulation that is based on the full characteristic matrices is also presented. The multidimensional Riemann solvers that are demonstrated here use one-dimensional HLLC Riemann solvers as building blocks.
Several stringent test problems drawn from hydrodynamics and MHD are presented to show that the method works. Results from structured and unstructured meshes demonstrate the versatility of our method. The reader is also invited to watch a video introduction to multidimensional Riemann solvers on http://www.nd.edu/~dbalsara/Numerical-PDE-Course.

View all citing articles on Scopus

View full text

Regular ArticleA Multidimensional Flux Function with Applications to the Euler and Navier-Stokes Equations

Abstract

Regular Article
A Multidimensional Flux Function with Applications to the Euler and Navier-Stokes Equations