Development of a turbulent boundary layer after a step from smooth to rough surface

The flow developing downstream of a step change from smooth to rough surface condition is studied in the light of Townsend’s wall similarity hypothesis. Previous studies seem to support the hypothesis for channel and pipe flows, but there are considerable controversies about its application to boundary layers and in particular to surface roughness formed by spanwise bars. It has been suggested that this controversy arises from insufficient separation of scales between the boundary layer thickness and the roughness length scale. An experimental investigation has therefore been undertaken where the flow evolves from a fully developed smooth wall boundary layer at high Reynolds numbers over a step in surface roughness (Re _θ = 13,400 at the step). The flow is mapped through the development of the internal layer until the flow is fully developed over the rough wall. The internal layer is found to grow as δ ∼ X ^0.73, and after about 15 boundary layer thicknesses at the step, the internal layer has reached the outer edge of the incoming layer. At the last rough wall measurement station, the Reynolds number has grown to Re _θ ≈ 32,600 and the ratio of boundary layer to roughness length scales is δ/k ≈ 140. The outer layer differences between the smooth and the rough wall data were found to be sufficiently small to conclude that for this setup the Townsend’s wall similarity hypothesis appears to hold.     

A comparison of some algorithms for the nonlinear least squares problem

The problem of minimizing a sum of squares of nonlinear functions is studied. To solve this problem one usually takes advantage of the fact that the objective function is of this special form. Doing this gives the Gauss-Newton method or modifications thereof. To study how these specialized methods compare with general purpose nonlinear optimization routines, test problems were generated where parameters determining the local behaviour of the algorithms could be controlled. The order of 1000 test problems were generated for testing three algorithms: the Gauss-Newton method, the Levenberg-Marquardt method and a quasi-Newton method.     

Analysis of hashing with chaining in the prime area

The expected performance of hashing with chaining in the prime area is analyzed. The method analyzed is briefly characterized as hashing with chaining of overflow records in the prime storage area, using one or several noncoalescing chains per bucket, and with random search for empty space. The analysis is asymptotic, and it is assumed that deletions do not occur. Simple closed formulas cannot be obtained, except in the case of bucket size one, but numerical results are readily computed. The expected performance compares favorably with that of other methods for handling overflow records.     

Inclusion of dynamic σ-π polarization in π-electronab initio calculations

Summary A method is presented, whereby dynamic - polarization, i.e. the correlation effect expressed by simultaneous (-*, -*) excitations, can be approximately included in a multi-reference configuration interaction (MRCI) or multi-configurational self-consistent field (MC-SCF) calculation, without need to explicitly correlate the sigma orbitals. The method, which we call the capacitance matrix method, is based on the use of conventional one-electron integrals, from which a polarization potential (SPP) contribution is computed and added to the one- and two-electron Hamiltonian. In the present form, the method requires one parameter for each type of atom, and one for each type of bond. These parameters were adjusted to reproduce the dynamic - polarization energy, computed by restricted multi-reference CI calculations, of a number of states of different hydrocarbons, and the agreement was within a few percent. Using the same parameters in CAS (Complete Active Space) SCF calculations of various states of benzene gives excitation energies, when SPP is included, which is comparable to those obtained by much more elaborate MRCI calculations.     

Perturbation theory for pseudo-inverses

A perturbation theory for pseudo-inverses is developed. The theory is based on a useful decomposition (theorem 2.1) ofB ⁺ -A ⁺ whereB andA arem ×n matrices. Sharp estimates of B ⁺ -A ⁺ are derived for unitary invariant norms whenA andB are of the same rank and B -A is small. Under similar conditions the perturbation of a linear systemAx=b is studied. Realistic bounds on the perturbation ofx=A ⁺ b andr=b=Ax are given. Finally it is seen thatA ⁺ andB ⁺ can be compared if and only ifR(A) andR(B) as well asR(A ^H ) andR(B ^H ) are in the acute case. Some theorems valid only in the acute case are also proved.This work was sponsored in part by The Swedish Institute of Applied Mathematics.     

Dynamic hashing

A new file organisation called dynamic hashing is presented. The organisation is based on normal hashing, but the allocated storage space can easily be increased and decreased without reorganising the file, according to the number of records actually stored in the file. The expected storage utilisation is analysed and is shown to be approximately 69% all the time. Algorithms for inserting and deleting a record are presented and analysed. Retrieval of a record is fast, requiring only one access to secondary storage. There are no overflow records. The proposed scheme necessitates maintenance of a relatively small index structured as a forest of binary trees or slightly modified binary tries. The expected size of the index is analysed and a compact representation of the index is suggested.     

Density matrix averaged atomic natural orbital (ANO) basis sets for correlated molecular wave functions

Summary Generally contracted basis sets for first row atoms have been constructed using the Atomic Natural Orbital (ANO) approach, with modifications for allowing symmetry breaking and state averaging. The ANOs are constructed by averaging over several atomic states, positive and negative ions, and atoms in an external electric field. The contracted basis sets give virtually identical results as the corresponding uncontracted sets for the atomic properties, which they have been designed to reproduce. The design objective has been to describe the ionization potential, the electron affinity, and the polarizability as accurately as possible. The result is a set of well-balanced basis sets for molecular calculations. The starting primitive sets are 8s4p3d for hydrogen, 9s4p3d for helium, and 14s9p4d3f for the heavier first row atoms.     

Dissociative recombination of HeH+. I. Rovibrational spectrum of HeH Rydberg states

The repulsive ground electronic state X²Σ⁺ of HeH is strongly coupled to the Rydberg states at small interatomic distances. Such large couplings also occur between some of the Rydberg states. HeH⁺ ions that capture an electron in a Rydberg state end up in separated He and H atoms by indirect predissociation. This paper presents a study of potential functions and pertinent matrix elements involving the lowest electronic states: the ²Σ⁺ states, X, A, C, and D, and the ²Π states B and E. Individual transition rates as well as total radiative and non-radiative lifetimes have been computed for the lowest vibrational and rotational levels. Received: 22 June 1998 / Accepted: 21 August 1998 / Published online: 12 October 1998     

Correlation potentials for a multiconfigurational-based density functional theory with exact exchange

A density functional theory based on a complete active space self consistent field (CASSCF) reference function with exact exchange is discussed. It is first shown that such a theory may be formulated with a correlation potential dependent on the density function and on the active space used. Auxiliary functions, such as the on-top two-particle density, are used to define uniquely the potential for different active spaces. The paper also analyses the correlation functional for some atomic and molecular cases. Large ab initio calculations are performed to obtain accurate density functions. A correlation potential is then fitted such that the reference CASSCF function gives the same density. The correlation potential values are saved in a data base for future analysis.     

A CASPT2 study of the valence and lowest Rydberg electronic states of benzene and phenol

Summary The valence excited states and the 3s, 3p, and 3d (united atom) Rydberg states of benzene and phenol have been obtained by the CASPT2 method, which computes a second-order perturbation correction to complete active space self-consistent field (CASSCF) energies. All non-zero dipole oscillator strengths are also computed, at the CASSCF level. For benzene, 16 singlet and 16 triplet states with excitation energies up to ca. 7.86 eV (63 400 cm^–1) are obtained. Of these, 12 singlet and three triplet energies are experimentally known well enough to allow meaningful comparison. The average error is around 0.1 eV. The highest of these singlet states (2¹ E_2g) is the highest valence * state predicted by elementary -electron theory. Its energy is then considerably lower than has been suggested from laser flash experiments, but in perfect agreement with a reinterpretation of that experiment. For phenol, 27 singlet states are obtained, in the range 4.53–7.84 eV (63 300 cm^–1). Only the lowest has a well-known experimental energy, which agrees with the computed result within 0.03 eV. The ionization energy is in error by 0.05 eV.