Research in computational fluid dynamics and analysis of algorithms

Recently, higher-order compact schemes have seen increasing use in the DNS (Direct Numerical Simulations) of the Navier-Stokes equations. Although they do not have the spatial resolution of spectral methods, they offer significant increases in accuracy over conventional second order methods. They can be used on any smooth grid, and do not have an overly restrictive CFL dependence as compared with the O(N(exp -2)) CFL dependence observed in Chebyshev spectral methods on finite domains. In addition, they are generally more robust and less costly than spectral methods. The issue of the relative cost of higher-order schemes (accuracy weighted against physical and numerical cost) is a far more complex issue, depending ultimately on what features of the solution are sought and how accurately they must be resolved. In any event, the further development of the underlying stability theory of these schemes is important. The approach of devising suitable boundary clusters and then testing them with various stability techniques (such as finding the norm) is entirely the wrong approach when dealing with high-order methods. Very seldom are high-order boundary closures stable, making them difficult to isolate. An alternative approach is to begin with a norm which satisfies all the stability criteria for the hyperbolic system, and look for the boundary closure forms which will match the norm exactly. This method was used recently by Strand to isolate stable boundary closure schemes for the explicit central fourth- and sixth-order schemes. The norm used was an energy norm mimicking the norm for the differential equations. Further research should be devoted to BC for high order schemes in order to make sure that the results obtained are reliable. The compact fourth order and sixth order finite difference scheme had been incorporated into a code to simulate flow past circular cylinders. This code will serve as a verification of the full spectral codes. A detailed stability analysis by Carpenter (from the fluid Mechanics Division) and Gottlieb gave analytic conditions for stability as well as asymptotic stability. This had been incorporated in the code in form of stable boundary conditions. Effects of the cylinder rotations had been studied. The results differ from the known theoretical results. We are in the middle of analyzing the results. A detailed analysis of the effects of the heating of the cylinder on the shedding frequency had been studied using the above schemes. It has been found that the shedding frequency decreases when the wire was heated. Experimental work is being carried out to affirm this result

Spurious frequencies as a result of numerical boundary treatments

The stability theory for finite difference Initial Boundary-Value approximations to systems of hyperbolic partial differential equations states that the exclusion of eigenvalues and generalized eigenvalues is a sufficient condition for stability. The theory, however, does not discuss the nature of numerical approximations in the presence of such eigenvalues. In fact, as was shown previously, for the problem of vortex shedding by a 2-D cylinder in subsonic flow, stating boundary conditions in terms of the primitive (non-characteristic) variables may lead to such eigenvalues, causing perturbations that decay slowly in space and remain periodic time. Characteristic formulation of the boundary conditions avoided this problem. A more systematic study of the behavior of the (linearized) one-dimensional gas dynamic equations under various sets of oscillation-inducing legal boundary conditions is reported

Convergence results for pseudospectral approximations of hyperbolic systems by a penalty type boundary treatment

A new method of imposing boundary conditions in the pseudospectral approximation of hyperbolic systems of equations is proposed. It is suggested to collocate the equations, not only at the inner grid points, but also at the boundary points and use the boundary conditions as penalty terms. In the pseudo-spectral Legrendre method with the new boundary treatment, a stability analysis for the case of a constant coefficient hyperbolic system is presented and error estimates are derived

Data mining of protein families using common peptides

Predicting the function of a protein from its sequence is typically addressed using sequence-similarity. Here we propose a motif-based approach, using supervised motif extraction from protein sequences belonging to one functional family. The resulting deterministic motifs form Common Peptides (CPs) that characterize this family, allow for data mining of its proteins and facilitate further partition of the family into cluster

Resolution properties of the Fourier method for discontinuous waves

In this paper we discuss the wave-resolution properties of the Fourier approximations of a wave function with discontinuities. It is well known that a minimum of two points per wave is needed to resolve a periodic wave function using Fourier expansions. For Chebyshev approximations of a wave function, a minimum of pi points per wave is needed. Here we obtain an estimate for the minimum number of points per wave to resolve a discontinuous wave based on its Fourier coefficients. In our recent work on overcoming the Gibbs phenomenon, we have shown that the Fourier coefficients of a discontinuous function contain enough information to reconstruct with exponential accuracy the coefficient of a rapidly converging Gegenbauer expansion. We therefore study the resolution properties of a Gegenbauer expansion where both the number of terms and the order increase

Parallel pseudospectral domain decomposition techniques

The influence of interface boundary conditions on the ability to parallelize pseudospectral multidomain algorithms is investigated. Using the properties of spectral expansions, a novel parallel two domain procedure is generalized to an arbitrary number of domains each of which can be solved on a separate processor. This interface boundary condition considerably simplifies influence matrix techniques

Spectral simulation of unsteady compressible flow past a circular cylinder

An unsteady compressible viscous wake flow past a circular cylinder was successfully simulated using spectral methods. A new approach in using the Chebyshev collocation method for periodic problems is introduced. It was further proved that the eigenvalues associated with the differentiation matrix are purely imaginary, reflecting the periodicity of the problem. It was been shown that the solution of a model problem has exponential growth in time if improper boundary conditions are used. A characteristic boundary condition, which is based on the characteristics of the Euler equations of gas dynamics, was derived for the spectral code. The primary vortex shedding frequency computed agrees well with the results in the literature for Mach = 0.4, Re = 80. No secondary frequency is observed in the power spectrum analysis of the pressure data

Splitting methods for low Mach number Euler and Navier-Stokes equations

Examined are some splitting techniques for low Mach number Euler flows. Shortcomings of some of the proposed methods are pointed out and an explanation for their inadequacy suggested. A symmetric splitting for both the Euler and Navier-Stokes equations is then presented which removes the stiffness of these equations when the Mach number is small. The splitting is shown to be stable

Terminal deoxynucleotidyltransferase. Serological studies and radioimmunoassay

Mouse antisera against calf terminal deoxynucleotidyltransferase (terminal transferase) have been prepared. The sera have been used to characterize terminal transferase both by studying inhibition of enzyme activity and by developing a competition radioimmunoassay using highly purified 125I-labeled terminal transferase. By either assay, anti-terminal transferase serum did not cross-react significantly with calf DNA polymerases alpha and beta, Escherichia coli DNA polymerase I, or the reverse transcriptase of Moloney mouse leukemia virus. The calf terminal transferase did, however, share cross-reactive but not identical determinants with human and murine terminal transferase. The radioimmunoassay could detect as little as 2 ng of terminal transferase/mg of soluble protein in a tissue extract. Thymocytes were found to contain 280 ng of terminal transferase/mg of cell protein or about 1 X 10^(5) molecules/cell; bone marrow had about 1% of the level of enzyme found in thymus. Extracts of spleen, peripheral white blood cells, lymph nodes, liver, muscle, and kidney all lacked detectable antigenicity of terminal transferase. These data indicate that terminal transferase is a tissue-specific enzyme and is not related to other DNA polymerases