835 research outputs found
Improvements to enhance robustness of third-order scale-independent WENO-Z schemes
Although there are many improvements to WENO3-Z that target the achievement
of optimal order in the occurrence of the first-order critical point (CP1),
they mainly address resolution performance, while the robustness of schemes is
of less concern and lacks understanding accordingly. In light of our analysis
considering the occurrence of critical points within grid intervals, we
theoretically prove that it is impossible for a scale-independent scheme that
has the stencil of WENO3-Z to fulfill the above order achievement, and current
scale-dependent improvements barely fulfill the job when CP1 occurs at the
middle of the grid cell. In order to achieve scale-independent improvements, we
devise new smoothness indicators that increase the error order from 2 to 4 when
CP1 occurs and perform more stably. Meanwhile, we construct a new global
smoothness indicator that increases the error order from 4 to 5 similarly,
through which new nonlinear weights with regard to WENO3-Z are derived and new
scale-independents improvements, namely WENO-ZES2 and -ZES3, are acquired.
Through 1D scalar and Euler tests, as well as 2D computations, in comparison
with typical scale-dependent improvement, the following performances of the
proposed schemes are demonstrated: The schemes can achieve third-order accuracy
at CP1 no matter its location in the stencil, indicate high resolution in
resolving flow subtleties, and manifest strong robustness in hypersonic
simulations (e.g., the accomplishment of computations on hypersonic
half-cylinder flow with Mach numbers reaching 16 and 19, respectively, as well
as essentially non-oscillatory solutions of inviscid sharp double cone flow at
M=9.59), which contrasts the comparative WENO3-Z improvement
An Adaptive Characteristic-wise Reconstruction WENOZ scheme for Gas Dynamic Euler Equations
Due to its excellent shock-capturing capability and high resolution, the WENO
scheme family has been widely used in varieties of compressive flow simulation.
However, for problems containing strong shocks and contact discontinuities,
such as the Lax shock tube problem, the WENO scheme still produces numerical
oscillations. To avoid such numerical oscillations, the characteristic-wise
construction method should be applied. Compared to component-wise
reconstruction, characteristic-wise reconstruction leads to much more
computational cost and thus is not suite for large scale simulation such as
direct numeric simulation of turbulence. In this paper, an adaptive
characteristic-wise reconstruction WENO scheme, i.e. the AdaWENO scheme, is
proposed to improve the computational efficiency of the characteristic-wise
reconstruction method. The new scheme performs characteristic-wise
reconstruction near discontinuities while switching to component-wise
reconstruction for smooth regions. Meanwhile, a new calculation strategy for
the WENO smoothness indicators is implemented to reduce over-all computational
cost. Several one dimensional and two dimensional numerical tests are performed
to validate and evaluate the AdaWENO scheme. Numerical results show that
AdaWENO maintains essentially non-oscillatory flow field near discontinuities
as the characteristic-wise reconstruction method. Besieds, compared to the
component-wise reconstruction, AdaWENO is about 40\% faster which indicates its
excellent efficiency
Low-diffusivity scalar transport using a WENO scheme and dual meshing
Interfacial mass transfer of low-diffusive substances in an unsteady flow
environment is marked by a very thin boundary layer at the interface and other
regions with steep concentration gradients. A numerical scheme capable of
resolving accurately most details of this process is presented. In this scheme,
the fourth-order accurate WENO method developed by Liu et al. (1994) was
implemented on a non-uniform staggered mesh to discretize the scalar convection
while for the scalar diffusion a fourth-order accurate central discretization
was employed. The discretization of the scalar convection-diffusion equation
was combined with a fourth-order Navier-Stokes solver which solves the
incompressible flow. A dual meshing strategy was employed, in which the scalar
was solved on a finer mesh than the incompressible flow. The solver was tested
by performing a number of two-dimensional simulations of an unstably stratified
flow with low diffusivity scalar transport. The unstable stratification led to
buoyant convection which was modelled using a Boussinesq approximation with a
linear relationship between flow temperature and density. The order of accuracy
for one-dimensional scalar transport on a stretched and uniform grid was also
tested. The results show that for the method presented above a relatively
coarse mesh is sufficient to accurately describe the fluid flow, while the use
of a refined mesh for the low-diffusive scalars is found to be beneficial in
order to obtain a highly accurate resolution with negligible numerical
diffusion
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