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NASA NTRS1988

An artificial compression method for ENO schemes: The slope modification method

A simple and effective method of artificial compression is introduced. This method is based on a modification of the slopes of the ENO (essentially nonoscillatory) reconstruction and, with the help of suitable chosen parameters, greatly improves the resolution of the contact discontinuities. Numerical examples are provided to test the performance of the method and to give some suggestions as to the choice of the parameters.

Yang, Huanan↗
NASA NTRS1991

Finite-volume application of high-order ENO schemes to two-dimensional boundary-value problems

Finite-volume applications of high-order accurate ENO schemes to two-dimensional boundary-value problems are studied. These schemes achieve high-order spatial accuracy, in smooth regions, by a piecewise polynomial approximation of the solution from cell averages. In addition, this spatial operation involves an adaptive stencil algorithm in order to avoid the oscillatory behavior that is associated with interpolation across steep gradients. High-order TVD Runge-Kutta methods are employed for time integration, thus making these schemes best suited for unsteady problems. Fifth- and sixth-order accurate applications are validated through a grid refinement study involving the solutions of scalar hyperbolic equations. A previously proposed extension for the Euler equations of gas dynamics is tested, including its application to solutions of boundary-value problems involving solid walls and curvilinear coordinates.

Casper, Jay↗
NASA NTRS1991

High-order ENO methods for the unsteady compressible Navier-Stokes equations

The adaptive stencil concepts of ENO (Essentially Non-Oscillatory) methods are applied to the laminar Navier-Stokes equations to yield a high-order, time-accurate algorithm with a shock-capturing capability. The method targets problems in the areas of nonlinear acoustics, compressible transition, and turbulence which, due to the presence of shocks or complex geometries, are not easily solved by spectral methods. The present approach has been implemented and tested for the full three-dimensional Navier-Stokes equations in a transformed curvilinear coordinate system. Validation results are presented for a variety of problems which verify the method's accuracy properties and shock capturing capabilities, as well as demonstrate its use as a direct simulation tool.

Atkins, H. L.↗
NASA NTRS1991

Multi-resolution analysis for ENO schemes

Given an function, u(x), which is represented by its cell-averages in cells which are formed by some unstructured grid, we show how to decompose the function into various scales of variation. This is done by considering a set of nested grids in which the given grid is the finest, and identifying in each locality the coarsest grid in the set from which u(x) can be recovered to a prescribed accuracy. This multi-resolution analysis was applied to essentially non-oscillatory (ENO) schemes in order to advance the solution by one time-step. This is accomplished by decomposing the numerical solution at the beginning of each time-step into levels of resolution, and performing the computation in each locality at the appropriate coarser grid. An efficient algorithm for implementing this program in the 1-D case is presented; this algorithm can be extended to the multi-dimensional case with Cartesian grids.

Harten, Ami↗
NASA NTRS1993

Multi-resolution analysis for ENO schemes

Given a function u(x) which is represented by its cell-averages in cells which are formed by some unstructured grid, we show how to decompose the function into various scales of variation. This is done by considering a set of nested grids in which the given grid is the finest, and identifying in each locality the coarsest grid in the set from which u(x) can be recovered to a prescribed accuracy. We apply this multi-resolution analysis to Essentially Non-oscillatory Schemes (ENO) schemes in order to reduce the number of numerical flux computations which is needed in order to advance the solution by one time-step. This is accomplished by decomposing the numerical solution at the beginning of each time-step into levels of resolution, and performing the computation in each locality at the appropriate coarser grid. We present an efficient algorithm for implementing this program in the one-dimensional case; this algorithm can be extended to the multi-dimensional case with cartesian grids.

Harten, Ami↗
NASA NTRS1990

An artificial compression method for ENO schemes - The slope modification method

A simple and effective method of artificial compression is introduced. This method is based on a modification of the slopes of the END (essentially nonoscillatory) reconstruction and, with the help of suitable chosen parameters, greatly improves the resolution of the contact discontinuities. Numerical examples are provided to test the performance of the method and to give some suggestions as to the choice of the parameters.

Yang, Huanan↗
NASA NTRS1993

A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems

The finite-volume approach is presently used to obtain a 2D, high-order accurate and basically nonoscillatory shock-capture method whose high-order spatial accuracy is obtained by means of a piecewise polynomial approximation of the solution from cell averages. Attention is given to a high-order spatial operator that is able to both retain high-order accuracy in smooth regions and avoid the oscillations that are associated with interpolations across steep gradients. The operator is extended to hyperbolic systems of equations and curvilinear meshes.

Casper, Jay↗
NASA NTRS2002

Shear stress stimulates phosphorylation of endothelial nitric-oxide synthase at Ser1179 by Akt-independent mechanisms: role of protein kinase A

Recently, we have shown that shear stress stimulates NO(*) production by the protein kinase B/Akt (Akt)-dependent mechanisms in bovine aortic endothelial cells (BAEC) (Go, Y. M., Boo, Y. C., Park, H., Maland, M. C., Patel, R., Pritchard, K. A., Jr., Fujio, Y., Walsh, K., Darley-Usmar, V., and Jo, H. (2001) J. Appl. Physiol. 91, 1574-1581). Akt has been believed to regulate shear-dependent production of NO(*) by directly phosphorylating endothelial nitric-oxide synthase (eNOS) at the Ser(1179) residue (eNOS-S(1179)), but a critical evaluation using specific inhibitors or dominant negative mutants (Akt(AA) or Akt(AAA)) has not been reported. In addition, other kinases, including protein kinase A (PKA) and AMP kinase have also shown to phosphorylate eNOS-S(1179). Here, we show that shear-dependent phosphorylation of eNOS-S(1179) is mediated by an Akt-independent, but a PKA-dependent, mechanism. Expression of Akt(AA) or Akt(AAA) in BAEC by using recombinant adenoviral constructs inhibited phosphorylation of eNOS-S(1179) if cells were stimulated by vascular endothelial growth factor (VEGF), but not by shear stress. As shown before, expression of Akt(AA) inhibited shear-dependent NO(*) production, suggesting that Akt is still an important regulator in NO production. Further studies showed that a selective inhibitor of PKA, H89, inhibited shear-dependent phosphorylation of eNOS-S(1179) and NO(*) production. In contrast, H89 did not inhibit phosphorylation of eNOS-S(1179) induced by expressing a constitutively active Akt mutant (Akt(Myr)) in BAEC, showing that the inhibitor did not affect the Akt pathway. 8-Bromo-cAMP alone phosphorylated eNOS-S(1179) within 5 min without activating Akt, in an H89-sensitive manner. Collectively, these results demonstrate that shear stimulates phosphorylation of eNOS-S(1179) in a PKA-dependent, but Aktindependent manner, whereas the NO(*) production is regulated by the mechanisms dependent on both PKA and Akt. A coordinated interaction between Akt and PKA may be an important mechanism by which eNOS activity is regulated in response to physiological stimuli such as shear stress.

Non-NASA Center↗
NASA NTRS1988

Efficient implementation of essentially non-oscillatory shock capturing schemes, 2

Earlier work on the efficient implementation of ENO (essentially non-oscillatory) shock capturing schemes is continued. A new simplified expression is provided for the ENO construction procedure based again on numerical fluxes rather than cell averages. Also considered are two improvements which are labeled ENO-LLF (local Lax-Friedrichs) and ENO-Roe, which yield sharper shock transitions, improved overall efficiency, and lower computational cost than previous implementation of the ENO schemes. Two methods of sharpening contact discontinuities, i.e., the subcell resolution idea of Harten and the artificial compression idea of Yang, which those authors used originally in the cell-average framework, are supplied to the current ENO schemes using numerical fluxes and TVD Runge-Kutta time discretizations. The implementation for nonlinear systems and multi-dimensions is given. Finally, many numerical examples, including a compressible shock turbulence interaction flow calculation, are given.

Shu, Chi-Wang↗
NASA NTRS1989

Efficient implementation of essentially non-oscillatory shock-capturing schemes. II

The present elaboration of essentially nonoscillatory (ENO) shock-capturing schemes proceeds with a novel, simplified expression for the ENO construction procedure having its basis in numerical fluxes rather than cell-averages. The ENO-local Lax-Friedrichs and ENO-Roe schemes introduced yield sharper shock transitions and greater overall accuracy than previous ENO-scheme implementations. Numerical fluxes and TVD Runge-Kutta time discretizations are used to apply Harten's subcell-resolution concept (1986) and Yang's artificial compression method to the current ENO schemes, in the interest of sharper contact discontinuities.

Shu, Chi-Wang↗
NASA NTRS1988

Essentially non-oscillatory shock capturing methods applied to turbulence amplification in shock wave calculations

ENO (essentially non-oscillatory) schemes can provide uniformly high order accuracy right up to discontinuities while keeping sharp, essentially non-oscillatory shock transitions. Recently, an efficient implementation of ENO schemes was obtained based on fluxes and TVD Runge-Kutta time discretizations. The resulting code is very simple to program for multi-dimensions. ENO schemes are especially suitable for computing problems with both discontinuities and fine structures in smooth regions, such as shock interaction with turbulence, for which results for 1-D and 2-D Euler equations are presented. Much better resolution is observed by using third order ENO schemes than by using second order TVD schemes for such problems.

Osher, Stanley↗
NASA NTRS1989

Recent progress on essentially non-oscillatory shock capturing schemes

An account is given of the construction of efficient implementations of 'essentially nonoscillatory' (ENO) schemes that approximate systems of hyperbolic conservation laws. ENO schemes use a local adaptive stencil to automatically obtain information from regions of smoothness when the solution develops discontinuities. Approximations employing ENOs can thereby obtain uniformly high accuracy to the very onset of discontinuities, while retaining a sharp and essentially nonoscillatory shock transition. For ease of implementation, ENO schemes applying the adaptive stencil concept to the numerical fluxes and employing a TVD Runge-Kutta-type time discretization are constructed.

Osher, Stanley↗
NASA NTRS1995

Numerical Study of Boundary Layer Interaction with Shocks: Method Improvement and Test Computation

The objective is the development of a high-order and high-resolution method for the direct numerical simulation of shock turbulent-boundary-layer interaction. Details concerning the spatial discretization of the convective terms can be found in Adams and Shariff (1995). The computer code based on this method as introduced in Adams (1994) was formulated in Cartesian coordinates and thus has been limited to simple rectangular domains. For more general two-dimensional geometries, as a compression corner, an extension to generalized coordinates is necessary. To keep the requirements or limitations for grid generation low, the extended formulation should allow for non-orthogonal grids. Still, for simplicity and cost efficiency, periodicity can be assumed in one cross-flow direction. For easy vectorization, the compact-ENO coupling algorithm as used in Adams (1994) treated whole planes normal to the derivative direction with the ENO scheme whenever at least one point of this plane satisfied the detection criterion. This is apparently too restrictive for more general geometries and more complex shock patterns. Here we introduce a localized compact-ENO coupling algorithm, which is efficient as long as the overall number of grid points treated by the ENO scheme is small compared to the total number of grid points. Validation and test computations with the final code are performed to assess the efficiency and suitability of the computer code for the problems of interest. We define a set of parameters where a direct numerical simulation of a turbulent boundary layer along a compression corner with reasonably fine resolution is affordable.

Adams, N. A.↗
NASA NTRS2003

Flow-dependent regulation of endothelial nitric oxide synthase: role of protein kinases

Vascular endothelial cells are directly and continuously exposed to fluid shear stress generated by blood flow. Shear stress regulates endothelial structure and function by controlling expression of mechanosensitive genes and production of vasoactive factors such as nitric oxide (NO). Though it is well known that shear stress stimulates NO production from endothelial nitric oxide synthase (eNOS), the underlying molecular mechanisms remain unclear and controversial. Shear-induced production of NO involves Ca2+/calmodulin-independent mechanisms, including phosphorylation of eNOS at several sites and its interaction with other proteins, including caveolin and heat shock protein-90. There have been conflicting results as to which protein kinases-protein kinase A, protein kinase B (Akt), other Ser/Thr protein kinases, or tyrosine kinases-are responsible for shear-dependent eNOS regulation. The functional significance of each phosphorylation site is still unclear. We have attempted to summarize the current status of understanding in shear-dependent eNOS regulation.

Non-NASA Center↗
NASA NTRS2003

Effects of simulated microgravity on arterial nitric oxide synthase and nitrate and nitrite content

The aim of the present work was to investigate the alterations in nitric oxide synthase (NOS) expression and nitrate and nitrite (NOx) content of different arteries from simulated microgravity rats. Male Wistar rats were randomly assigned to either a control group or simulated microgravity group. For simulating microgravity, animals were subjected to hindlimb unweighting (HU) for 20 days. Different arterial tissues were removed for determination of NOS expression and NOx. Western blotting was used to measure endothelial NOS (eNOS) and inducible NOS (iNOS) protein content. Total concentrations of NOx, stable metabolites of nitric oxide, were determined by the chemiluminescence method. Compared with controls, isolated vessels from simulated microgravity rats showed a significant increase in both eNOS and iNOS expression in carotid arteries and thoracic aorta and a significant decrease in eNOS and iNOS expression of mesenteric arteries. The eNOS and iNOS content of cerebral arteries, as well as that of femoral arteries, showed no differences between the two groups. Concerning NOx, vessels from HU rats showed an increase in cerebral arteries, a decrease in mesenteric arteries, and no change in carotid artery, femoral artery and thoracic aorta. These data indicated that there were differential alterations in NOS expression and NOx of different arteries after hindlimb unweighting. We suggest that these changes might represent both localized adaptations to differential body fluid redistribution and other factors independent of hemodynamic shifts during simulated microgravity.

Non-NASA Center↗
NASA NTRS1990

High order filtering methods for approximating hyberbolic systems of conservation laws

In the computation of discontinuous solutions of hyperbolic systems of conservation laws, the recently developed essentially non-oscillatory (ENO) schemes appear to be very useful. However, they are computationally costly compared to simple central difference methods. A filtering method which is developed uses simple central differencing of arbitrarily high order accuracy, except when a novel local test indicates the development of spurious oscillations. At these points, the full ENO apparatus is used, maintaining the high order of accuracy, but removing spurious oscillations. Numerical results indicate the success of the method. High order of accuracy was obtained in regions of smooth flow without spurious oscillations for a wide range of problems and a significant speed up of generally a factor of almost three over the full ENO method.

Lafon, F.↗
NASA NTRS1991

High order filtering methods for approximating hyperbolic systems of conservation laws

The essentially nonoscillatory (ENO) schemes, while potentially useful in the computation of discontinuous solutions of hyperbolic conservation-law systems, are computationally costly relative to simple central-difference methods. A filtering technique is presented which employs central differencing of arbitrarily high-order accuracy except where a local test detects the presence of spurious oscillations and calls upon the full ENO apparatus to remove them. A factor-of-three speedup is thus obtained over the full-ENO method for a wide range of problems, with high-order accuracy in regions of smooth flow.

Lafon, F.↗
NASA NTRS1992

A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow

High order essentially non-oscillatory (ENO) schemes, originally designed for compressible flow and in general for hyperbolic conservation laws, are applied to incompressible Euler and Navier-Stokes equations with periodic boundary conditions. The projection to divergence-free velocity fields is achieved by fourth order central differences through Fast Fourier Transforms (FFT) and a mild high-order filtering. The objective of this work is to assess the resolution of ENO schemes for large scale features of the flow when a coarse grid is used and small scale features of the flow, such as shears and roll-ups, are not fully resolved. It is found that high-order ENO schemes remain stable under such situations and quantities related to large-scale features, such as the total circulation around the roll-up region, are adequately resolved.

Weinan, E.↗