TENO scheme

In this thesis, several novel numerical methods, including the high-order shock-capturing targeted Essentially Non-oscillatory (TENO) schemes, smoothed-particle hydrodynamics (SPH) based partitioning method and Centroidal Voronoi Particle (CVP) domain decomposition method, are proposed and validated. These methods can be used for high-resolution computational fluid simulations and high-performance scientific computing, which are important to study the physics of fluid dynamics.

With the observation that the TENO weighting strategy can explicitly distinguish smooth scales from nonsmooth scales in spectral space, in this paper, a new discontinuity indicator is proposed based on the high-order TENO paradigm [Fu et al., JCP 305(2016): 333-359]. The local flow structures are classified as smooth or non-smooth scales, and the hybrid numerical discretization scheme is applied correspondingly , i.e. the high-order upwind linear scheme without characteristic decomposition is employed for resolving smooth scales while the nonlinear low-dissipation TENO scheme is adopted to capture discontinuities. Since the time-consuming characteristic decomposition and smoothness-indicator computation of TENO are avoided in smooth regions, the overall computational efficiency can be improved significantly. Moreover, the cutoff wavenumber separating smooth and nonsmooth scales is determined by the parameter C T. In contrast to the thresholds of other discontinuity indicators , which are typically defined in physical space, C T takes effects in wavespace rendering its high generality. A set of benchmark cases with widespread length-scales is simulated to assess the performance of the proposed discontinuity indicator and the resulting hybrid shock-capturing scheme. Compared to the monotonicity-preserving discontinuity indicator and the TVB discontinuity indicator, the proposed algorithm delivers better performance with a fixed set of parameters for all considered benchmarks .

In [Fu et al., JCP 305(2016): 333-359], a family of high-order targeted ENO (TENO) schemes with a new nonlinear weighting strategy has been proposed. Building upon this strategy, in the current paper, a new class of adaptive TENO schemes for hyperbolic conservation laws is proposed based on three new concepts: (1) a hierarchical voting strategy is proposed to improve the ENO-like stencil selection; (2) the TENO weighting strategy is extended to function as a built-in discontinuity-location detector. Since the reconstruction scheme must not cross any discontinuity, corresponding target schemes are selected from a set of predefined linear schemes which are optimized towards maximum accuracy order or spectral resolution; (3) based on the observation that the cut-off parameter $C_T$ in the TENO weighting strategy determines nonlinear dissipation, a $C_T$ adaptation strategy is developed to minimize numerical dissipation for high-wavenumber fluctuations while maintaining robustness for shock-capturing. Six-point and eight-point TENO schemes are constructed, and their spectral properties are analyzed by the ADR analysis. A set of benchmark cases is considered to demonstrate the performance of proposed adaptive TENO schemes. Numerical results suggest that the proposed TENO schemes preserve the accuracy order at first and second order critical points and show less numerical dissipation compared with typical WENO schemes. Moreover, the TENO8-NA scheme exhibits good results for both highly compressible flows and nearly incompressible turbulence.

In [Fu et al., JCP 305(2016): 333-359], a family of high-order targeted ENO (TENO) schemes is proposed. The weighting strategy of TENO either applies a candidate stencil with its optimal weight, or removes its contribution completely when it is crossed by discontinuities. This ENO-like stencil selection procedure significantly diminishes the numerical dissipation induced by the nonlinear adaptations of classical WENO schemes. In this paper, the fifth-order TENO scheme is extended to simulate re-active flows in combination with an uncoupled method [1, 2], which splits the reaction source term of detailed chemistry from the flow equations. A set of benchmark cases including the two-dimensional self-sustained detonation is simulated to validate and compare the performance of the fifth-order WENO and TENO schemes. Numerical experiments demonstrate that TENO scheme is robust for simulating chemical reacting flows with using the uncoupled method. In particular, TENO scheme shows better performance in capturing both the shockwaves and the small-scale flow structures, e.g. shear layers and vortices.

In Fu et al., (2016), a family of high-order TENO shock-capturing schemes has been proposed for compressible fluid simulations within a finite-difference framework. With the TENO weighting strategy, each candidate stencil is either applied for the final reconstruction with its optimal weight or discarded completely when crossed by discontinuities. In this paper, with the observation that the local flow scales can be judged to be smooth or non-smooth explicitly, we propose a novel low-dissipation finite-volume method based on a new TENO reconstruction. Firstly, a new ENO-like stencil selection paradigm, which adapts between three three-point small stencils and a large candidate stencil, is proposed. The resulting TENO scheme inherits the low-dissipation advantage of original TENO schemes and can be extended to arbitrarily high-order reconstruction without significant complexity increase. The optimal background linear scheme on the three small stencils and that on the large stencil can be optimized either approaching high-order accuracy or better spectral properties separately. Secondly, within the finite-volume framework, a ”low-dissipation” Riemann solver is applied for flux computing when the large candidate stencils for both the left- and right- side reconstruction are judged as smooth whereas a robust ”dissipative” Riemann solver is adopted when one large candidate stencil crosses discontinuities. Since the numerical dissipation from both the reconstruction stage and the flux computing stage can be tuned according to the TENO weighting strategy, the proposed finite-volume method is less-dissipative and provides additional flexibility to handle challenging simulations. A set of benchmark cases is simulated to assess the performance of proposed method.

In this paper, we propose an improvement of the five-point and six-point targeted ENO schemes [Fu et al., JCP 305 (2016): 333-359] by introducing an adaptive dissipation control strategy. Nonlinear numerical dissipation is controlled by dynamically adjusting the cut-off parameter in the TENO weighting strategy according to the first-order smoothness measure of the local flow scales. For the five-point reconstruction, the dissipation bandwidth of the underlying linear scheme is delayed by introducing slight anti-dissipation at low wavenumbers with small dispersion errors. A new sixth order scale-separation parameter τ5 is derived and the modified TENO5-A scheme is third-order accurate. For the six-point reconstruction, dispersion and dissipation error are optimized separately resulting in a modified fourth-order TENO6-A scheme. All necessary parameters are determined by spectral analyses and are shown to be problem-independent by numerical experiments. A set of benchmark cases, including highly compressible gas dynamics, nearly incompressible and compressible turbulence decay, is considered. Numerical experiments demonstrate that both the proposed TENO5-A and TENO6-A scheme show excellent performance for shocks and broadband turbulence. The turbulence statistics obtained with TENO6-A at coarse resolution are comparable to those from the state-of-the-art implicit LES model and agree well with DNS data.

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In the future, very-high-Mach-number vehicles flying at low altitudes could play an important role in global transportation (Urzay 2018). By flying at relatively low altitudes, these systems can use ambient atmospheric oxygen for combustion, instead of conventional rockets, which carry their oxidizer in cryogenic tanks, the weight of which substantially reduces the efficiency of such transportation systems. While the thicker atmosphere is good for air-breathing propulsion, it leads to immense aerodynamic heating, which necessitates accurate numerical simulation to design safe and reliable systems. High-speed flows are numerically challenging to simulate due to the formation of shock waves, which are discontinuities in the continuous governing equations (Shu 2009; Pirozzoli 2011; Fu et al. 2021; Griffin et al. 2021), which are governed by hyperbolic conservation laws. In turbulent flows, it is also essential that the methods have very low numerical dissipation as not to artificially suppress turbulence. In addition, numerical robustness, which is the property that the solution remains bounded (does not blow up), is essential. Since the 1980s, many numerical approaches have been developed, such as the artificial viscosity scheme (Von & Richtmyer 1950; Jameson 1994), total variation diminishing scheme (Harten 1983), essentially non-oscillatory (ENO) (Harten et al. 1987) scheme and weighted essentially non-oscillatory (WENO) scheme (Liu et al. 1994). Among these approaches, the WENO scheme has been widely used in both academic and engineering simulations due to the sharp and robust shock-capturing capabilities. However, there are still some drawbacks, e.g., the order-degeneration near critical points, generating excessive numerical dissipation for small-scale structures, and the lack of numerical robustness for very-high-order reconstructions. Improvements for the order degeneration problem include the WENO-M scheme (Henrick et al. 2005) and the WENO-Z scheme (Borges et al. 2008). The numerical dissipation can be reduced by increasing the order of accuracy (Gerolymos et al. 2009), switching between a low-dissipation and a shock-capturing scheme based on a shock senor (the hybrid concept) (Adams & Shariff 1996), and optimizing the spectral properties of the scheme (improving the modified wavenumber) (Lele 1992; Weirs & Candler 1997). For enhancing the robustness of the scheme, Suresh & Huynhb (1997) propose to bound the solution near discontinuities by a monotonicitypreserving limiter. The order-reduction approach (Gerolymos et al. 2009; Titarev & Toro 2004), which recursively resorts to lower-order stencils, is an alternative way to improve the robustness of WENO schemes. However, when the above mentioned numerical approaches are applied to very-high-Mach-number flows with vacuum or near-vacuum regions, e.g., for applications with low density and low pressure, numerical difficulties may occur due to the presence of negative density and pressure. Although these high-order shock-capturing schemes generate entropy solutions for hyperbolic conservation laws (i.e., entropy monotonically increases), in general, they may fail to preserve positivity, which eventually causes the numerical

Accurate prediction of aerothermal surface loading is of paramount importance for the design of high-speed flight vehicles. In this work, we consider the numerical solution of hypersonic flow over a double-finned geometry, representative of the inlet of an air-breathing flight vehicle, characterized by threedimensional intersecting shock-wave/turbulent boundary layer interaction at Mach 8.3. High Reynolds numbers (Re L ≈ 11.6 × 10 6 based on free-stream conditions) and the presence of cold walls (T w /T • ≈ 0.26) leading to large near-wall temperature gradients necessitate the use of wall-modeled large eddy simulation (WMLES) in order to make calculations computationally tractable. The comparison of the WMLES results with experimental measurements shows good agreement in the time-averaged surface heat flux and wall pressure distributions, and the WMLES predictions show reduced errors with respect to the experimental measurements than prior RANS calculations. The favorable comparisons are obtained using a standard LES wall model based on equilibrium boundary layer approximations despite the presence of numerous non-equilibrium conditions including threedimensionality in the mean, shock/boundary layer interactions, and flow separation. We demonstrate that the use of semi-local eddy viscosity scaling (in lieu of the commonly used van Driest scaling) in the LES wall model is necessary to accurately predict the surface pressure loading and heat fluxes.

In this paper, an efficient low-dissipation high-order TENO scheme is proposed for ideal MHD flows. For high computational efficiency, a troubled-cell indicator based on the ENOlike stencil selection strategy in TENO schemes is introduced to isolate the discontinuities from smooth regions. While the high-order linear scheme is adopted for the smooth regions, a low-dissipation TENO scheme is applied for capturing discontinuities detected by the troubled-cell indicator. The case-independent parameters are given based on spectral analysis. Both the governing equations of the ideal MHD and the Hamilton-Jacobi type constrained transport equation for divergence-free condition can be solved by the newly proposed scheme. Since most computational regions are resolved by the linear scheme without expensive characteristic decomposition, flux splitting and nonlinear weight calculation, the proposed scheme is highly efficient. A set of benchmark cases has been simulated to demonstrate the performance of the proposed scheme. Numerical results reveal that remarkable speedup is achieved by the present scheme while the oscillation-free property and the high-order accuracy are preserved.