Recent developments in accuracy and stability improvement of nonlinear filter methods for DNS and LES of compressible flows

Recent progress in the improvement of numerical stability and accuracy of the Yee and Sjögreen [49] high order nonlinear filter schemes is described. The Yee & Sjögreen adaptive nonlinear filter method consists of a high order non-dissipative spatial base scheme and a nonlinear filter step. The nonlinear filter step consists of a flow sensor and the dissipative portion of a high resolution nonlinear high order shock-capturing method to guide the application of the shock-capturing dissipation where needed. The nonlinear filter idea was first initiated by Yee et al. [54] using an artificial compression method (ACM) of Harten [12] as the flow sensor. The nonlinear filter step was developed to replace high order linear filters so that the same scheme can be used for long time integration of direct numerical simulations (DNS) and large eddy simulations (LES) for both shock-free turbulence and turbulence-shock waves inter- actions. The improvement includes four major new developments: (a) Smart flow sensors were developed to replace the global ACM flow sensor [21,22,50]. The smart flow sensor provides the locations and the estimated strength of the necessary numerical dissipation needed at these locations and leaves the rest of the flow field free of shock-capturing dissipation. (b) Skew-symmetric splittings were developed for compressible gas dynamics and magnetohydrodynamics (MHD) equations [35,36] to improve numerical stability for long time integration. (c) High order entropy stable numerical fluxes were developed as the spatial base schemes for both the compressible gas dynamics and MHD [37,38]. (d) Several dispersion relation-preserving (DRP) central spatial schemes were included as spatial base schemes in the frame- work of our nonlinear filter method approach [40]. With these new scheme constructions the nonlinear filter schemes are applicable to a wider class of accurate and stable DNS and LES applications, including forced turbulence simulations where the time evolution of flows might start with low speed shock-free turbulence and develop into supersonic speeds with shocks. Representative test cases for both smooth flows and problems containing discontinuities for compressible flows are included

Development of low dissipative high order filter schemes for multiscale Navier–Stokes/MHD systems

Recent progress in the development of a class of low dissipative high order (fourth-order or higher) filter schemes for multiscale Navier–Stokes, and ideal and non-ideal magnetohydrodynamics (MHD) systems is described. The four main features of this class of schemes are: (a) multiresolution wavelet decomposition of the computed flow data as sensors for adaptive numerical dissipative control, (b) multistep filter to accommodate efficient application of different numerical dissipation models and different spatial high order base schemes, (c) a unique idea in solving the ideal conservative MHD system (a non-strictly hyperbolic conservation law) without having to deal with an incomplete eigensystem set while at the same time ensuring that correct shock speeds and locations are computed, and (d) minimization of the divergence of the magnetic field numerical error. By design, the flow sensors, different choice of high order base schemes and numerical dissipation models are stand-alone modules. A whole class of low dissipative high order schemes can be derived at ease, making the resulting computer software very flexible with widely applicable. Performance of multiscale and multiphysics test cases are illustrated with many levels of grid refinement and comparison with commonly used schemes in the literature

Accuracy consideration by DRP schemes for DNS and LES of compressible flow computations

Several dispersion relation-preserving (DRP) spatially central discretizations are considered as the base scheme in the framework of the Yee & Sjögreen low dissipative nonlinear filter approach. In addition, the nonlinear filter of Yee & Sjögreen with shock-capturing and long time integration capabilities is used to replace the standard DRP linear filter for both smooth flows and flows containing discontinuities. DRP schemes for computational aeroacoustics (CAA) focus on dispersion error consideration for long time lin- ear wave propagation rather than the formal order of accuracy of the scheme. The resulting DRP schemes usually have wider grid stencils and increased CPU operations count compared with standard central schemes of the same formal order of accuracy. For discontinuous initial data and long time wave propa- gation of smooth acoustic waves, various space and time DRP linear filter are needed. For acoustic waves interacting with shocks and turbulence induced noise, DRP schemes with linear filters alone usually are not capable of simulating such flows. The investigation presented in this paper is focused on the pos- sible gain in efficiency and accuracy by spatial DRP schemes over standard central schemes having the same grid stencil width for general direct numerical simulations (DNS) and large eddy simulations (LES) of compressible flows. Representative test cases for both smooth flows and problems containing discontinuities for 3D DNS of compressible gas dynamics are included

Grid convergence of high order methods for multiscale complex unsteady viscous compressible flows

Grid convergence of several high order methods for the computation of rapidly developing complex unsteady viscous compressible flows with a wide range of physical scales is studied. The recently developed adaptive numerical dissipation control high order methods referred to as the ACM and wavelet filter schemes are compared with a fifth-order weighted ENO (WENO) scheme. The two 2-D compressible full Navier–Stokes models considered do not possess known analytical and experimental data. Fine grid solutions from a standard second-order TVD scheme and a MUSCL scheme with limiters are used as reference solutions. The first model is a 2-D viscous analog of a shock tube problem which involves complex shock/shear/boundary-layer interactions. The second model is a supersonic reactive flow concerning fuel breakup. The fuel mixing involves circular hydrogen bubbles in air interacting with a planar moving shock wave. Both models contain fine scale structures and are stiff in the sense that even though the unsteadiness of the flows are rapidly developing, extreme grid refinement and time step restrictions are needed to resolve all the flow scales as well as the chemical reaction scales. Our computations were all made on uniform grids, and our conclusions cannot be directly carried over to, for example, curvilinear grids

Multiway real-time PCR gene expression profiling in yeast Saccharomyces cerevisiae reveals altered transcriptional response of ADH-genes to glucose stimuli

<p>Abstract</p> <p>Background</p> <p>The large sensitivity, high reproducibility and essentially unlimited dynamic range of real-time PCR to measure gene expression in complex samples provides the opportunity for powerful multivariate and multiway studies of biological phenomena. In multiway studies samples are characterized by their expression profiles to monitor changes over time, effect of treatment, drug dosage etc. Here we perform a multiway study of the temporal response of four yeast <it>Saccharomyces cerevisiae </it>strains with different glucose uptake rates upon altered metabolic conditions.</p> <p>Results</p> <p>We measured the expression of 18 genes as function of time after addition of glucose to four strains of yeast grown in ethanol. The data are analyzed by matrix-augmented PCA, which is a generalization of PCA for 3-way data, and the results are confirmed by hierarchical clustering and clustering by Kohonen self-organizing map. Our approach identifies gene groups that respond similarly to the change of nutrient, and genes that behave differently in mutant strains. Of particular interest is our finding that <it>ADH4 </it>and <it>ADH6 </it>show a behavior typical of glucose-induced genes, while <it>ADH3 </it>and <it>ADH5 </it>are repressed after glucose addition.</p> <p>Conclusion</p> <p>Multiway real-time PCR gene expression profiling is a powerful technique which can be utilized to characterize functions of new genes by, for example, comparing their temporal response after perturbation in different genetic variants of the studied subject. The technique also identifies genes that show perturbed expression in specific strains.</p

Recent developments in accuracy and stability improvement of nonlinear filter methods for DNS and LES of compressible flows

Recent progress in the improvement of numerical stability and accuracy of the Yee and Sjögreen [49] high order nonlinear filter schemes is described. The Yee & Sjögreen adaptive nonlinear filter method consists of a high order non-dissipative spatial base scheme and a nonlinear filter step. The nonlinear filter step consists of a flow sensor and the dissipative portion of a high resolution nonlinear high order shock-capturing method to guide the application of the shock-capturing dissipation where needed. The nonlinear filter idea was first initiated by Yee et al. [54] using an artificial compression method (ACM) of Harten [12] as the flow sensor. The nonlinear filter step was developed to replace high order linear filters so that the same scheme can be used for long time integration of direct numerical simulations (DNS) and large eddy simulations (LES) for both shock-free turbulence and turbulence-shock waves inter- actions. The improvement includes four major new developments: (a) Smart flow sensors were developed to replace the global ACM flow sensor [21,22,50]. The smart flow sensor provides the locations and the estimated strength of the necessary numerical dissipation needed at these locations and leaves the rest of the flow field free of shock-capturing dissipation. (b) Skew-symmetric splittings were developed for compressible gas dynamics and magnetohydrodynamics (MHD) equations [35,36] to improve numerical stability for long time integration. (c) High order entropy stable numerical fluxes were developed as the spatial base schemes for both the compressible gas dynamics and MHD [37,38]. (d) Several dispersion relation-preserving (DRP) central spatial schemes were included as spatial base schemes in the frame- work of our nonlinear filter method approach [40]. With these new scheme constructions the nonlinear filter schemes are applicable to a wider class of accurate and stable DNS and LES applications, including forced turbulence simulations where the time evolution of flows might start with low speed shock-free turbulence and develop into supersonic speeds with shocks. Representative test cases for both smooth flows and problems containing discontinuities for compressible flows are included

Development of low dissipative high order filter schemes for multiscale Navier–Stokes/MHD systems

Recent progress in the development of a class of low dissipative high order (fourth-order or higher) filter schemes for multiscale Navier–Stokes, and ideal and non-ideal magnetohydrodynamics (MHD) systems is described. The four main features of this class of schemes are: (a) multiresolution wavelet decomposition of the computed flow data as sensors for adaptive numerical dissipative control, (b) multistep filter to accommodate efficient application of different numerical dissipation models and different spatial high order base schemes, (c) a unique idea in solving the ideal conservative MHD system (a non-strictly hyperbolic conservation law) without having to deal with an incomplete eigensystem set while at the same time ensuring that correct shock speeds and locations are computed, and (d) minimization of the divergence of the magnetic field numerical error. By design, the flow sensors, different choice of high order base schemes and numerical dissipation models are stand-alone modules. A whole class of low dissipative high order schemes can be derived at ease, making the resulting computer software very flexible with widely applicable. Performance of multiscale and multiphysics test cases are illustrated with many levels of grid refinement and comparison with commonly used schemes in the literature

Grid convergence of high order methods for multiscale complex unsteady viscous compressible flows

Grid convergence of several high order methods for the computation of rapidly developing complex unsteady viscous compressible flows with a wide range of physical scales is studied. The recently developed adaptive numerical dissipation control high order methods referred to as the ACM and wavelet filter schemes are compared with a fifth-order weighted ENO (WENO) scheme. The two 2-D compressible full Navier–Stokes models considered do not possess known analytical and experimental data. Fine grid solutions from a standard second-order TVD scheme and a MUSCL scheme with limiters are used as reference solutions. The first model is a 2-D viscous analog of a shock tube problem which involves complex shock/shear/boundary-layer interactions. The second model is a supersonic reactive flow concerning fuel breakup. The fuel mixing involves circular hydrogen bubbles in air interacting with a planar moving shock wave. Both models contain fine scale structures and are stiff in the sense that even though the unsteadiness of the flows are rapidly developing, extreme grid refinement and time step restrictions are needed to resolve all the flow scales as well as the chemical reaction scales. Our computations were all made on uniform grids, and our conclusions cannot be directly carried over to, for example, curvilinear grids

Accuracy consideration by DRP schemes for DNS and LES of compressible flow computations

Several dispersion relation-preserving (DRP) spatially central discretizations are considered as the base scheme in the framework of the Yee & Sjögreen low dissipative nonlinear filter approach. In addition, the nonlinear filter of Yee & Sjögreen with shock-capturing and long time integration capabilities is used to replace the standard DRP linear filter for both smooth flows and flows containing discontinuities. DRP schemes for computational aeroacoustics (CAA) focus on dispersion error consideration for long time lin- ear wave propagation rather than the formal order of accuracy of the scheme. The resulting DRP schemes usually have wider grid stencils and increased CPU operations count compared with standard central schemes of the same formal order of accuracy. For discontinuous initial data and long time wave propa- gation of smooth acoustic waves, various space and time DRP linear filter are needed. For acoustic waves interacting with shocks and turbulence induced noise, DRP schemes with linear filters alone usually are not capable of simulating such flows. The investigation presented in this paper is focused on the pos- sible gain in efficiency and accuracy by spatial DRP schemes over standard central schemes having the same grid stencil width for general direct numerical simulations (DNS) and large eddy simulations (LES) of compressible flows. Representative test cases for both smooth flows and problems containing discontinuities for 3D DNS of compressible gas dynamics are included