A continuously differentiable upwinding scheme for the simulation of fluid flow problems

A C2‐continuous high‐resolution upwind convection scheme

“…The emphasis of this article are the following: (i) to introduce a simple upwinding scheme for the treatment of (linear and nonlinear) convective terms, with particular care to ensure boundedness and robustness; (ii) to perform a comparative study of the ADBQUICKEST, CUBISTA, WACEB, TOPUS, SDPUS‐C1, and EPUS schemes to solve standard conservation laws; and (iii) to document results for complex fluid dynamics problems encountered in practical engineering applications. In particular, this study can be regarded as an extension to that by Lima et al and a generalization of Lin and Chieng for numerical approximation of convection terms. The selected test cases are the following: (i) advection of scalars; (ii) interaction of colliding shock waves in gases; (iii) circular hydraulic jump in a Newtonian fluid; (iv) Newtonian and non‐Newtonian lid‐driven cavity flows; and (v) a gas‐solid flow in a bubbling fluidized bed.…”

“…According to Leonard , a bounded second and/or third‐order accurate upwinding scheme within the CBC region of Gaskell and Lau must pass through points O(0, 0), Q(0.5, 0.75), P(1, 1) and with inclination of 0.75 at Q (passing through Q will provide second‐order accuracy and passing through Q with a slope of 0.75 will give third‐order accuracy). Following a similar procedure to that used in Lima et al , the EPUS scheme is derived by assuming that, for , the (normalized) convected variable ϕ at a cell interface f (the interface g follows the same procedure) is an eight‐degree polynomial function with coefficients a k , k = 0,1,...,8, and for , it corresponds to the first‐order upwind scheme . Fixing a 3 as a free parameter (here denoted as λ ), the other coefficients are determined by imposing the Leonard's conditions together with the (C 2 ) continuously differentiable conditions and .…”

“…The main steps of the numerical methodology used (the Godunov method plus a correction term) for the 1‐D case can be found in Lima et al . The EPUS flux limiter is implemented in a specific routine of the CLAWPACK by setting it in Equation of . The time step, in this case, is assumed to satisfy the restriction where are the numerical acoustic waves associated with the numerical flux function.…”

“…Both generalized simplified marker‐and‐cell and EPUS techniques have been incorporated into the Freeflow code of Castelo et al for 2‐D and axisymmetric Newtonian and non‐Newtonian fluid flows. The implementation of the EPUS scheme follows a similar procedure as that in Section 3.2 of , with the convected variables assuming one of the components u or v in Equations , or the non‐Newtonian stress tensor components Γ ij in Equation . For instance, the convective terms in Equation with α = 0 are approximated by the conservative scheme where the advection velocities , , , and are obtained by averaging.…”

“…A major drawback of these latter schemes is that they are not well adapted to the simulation of problems of greater complexity such as viscoelastic, free surface, or gas-solid two-phase flows; their calculations are not based on functions of class C 2 , but on piecewise linear ones, like performed in [7].Hence, upwinding using sufficiently smooth functions (say polynomials) constitutes a promising option to search for a robust approximation for general convection terms. Within this scenario, two convection boundedness criterion/total variation diminishing (CBC [8] /TVD [9])-based upwind schemes have recently been introduced, namely third-order polynomial upwind scheme (TOPUS)[10] and six-degree polynomial upwind scheme of class C 1 (SDPUS-C1) [11], whose authors have shown that these polynomial schemes are able to handle several fluid dynamic's problems. However, both TOPUS and SDPUS-C1 schemes are not developed on the basis of class C 2 functions; TOPUS is not continuously differentiable, because near points (0,0) and (1,1) into the normalized diagram of Leonard [12] it is not smooth whereas SDPUS-C1 is a function of class C 1 only.…”