C. David Pruett

Spatial direct numerical simulation of high-speed boundary-layer flows part I: Algorithmic considerations and validation

A highly accurate algorithm for the direct numerical simulation (DNS) of spatially evolving high-speed boundary-layer flows is described in detail and is carefully validated. To represent the evolution of instability waves faithfully, the fully explicit scheme relies on non-dissipative high-order compact-difference and spectral collocation methods. Several physical, mathematical, and practical issues relevant to the simulation of high-speed transitional flows are discussed. In particular, careful attention is paid to the implementation of inflow, outflow, and far-field boundary conditions. Four validation cases are presented, in which comparisons are made between DNS results and results obtained from either compressible linear stability theory or from the parabolized stability equation (PSE) method, the latter of which is valid for nonparallel flows and moderately nonlinear disturbance amplitudes. The first three test cases consider the propagation of two-dimensional second-mode disturbances in Mach 4.5 flat-plate boundary-layer flows. The final test case considers the evolution of a pair of oblique second-mode disturbances in a Mach 6.8 flow along a sharp cone. The agreement between the fundamentally different PSE and DNS approaches is remarkable for the test cases presented.

Spatial direct numerical simulation of high-speed boundary-layer flows Part II: Transition on a cone in Mach 8 flow

The laminar breakdown of the boundary-layer flow of an axisymmetric sharp cone in a Mach 8 flow is simulated by a synergistic approach that combines the parabolized stability equation (PSE) method and spatial direct numerical simulation (DNS). The transitional state is triggered by a symmetric pair of oblique second-mode disturbances whose nonlinear interactions generate strong streamwise vorticity, which leads in turn to severe spanwise variations in the flow and eventual laminar breakdown. The PSE method is used to compute the weakly and moderately nonlinear initial stages of the transition process and, thereby, to derive a harmonically rich inflow condition for the DNS. The strongly nonlinear and laminar-breakdown stages of transition are then computed by well-resolved DNS, with a highly accurate algorithm that exploits spectral collocation and high-order compact-difference methods. Evolution of the flow is presented in terms of modal energies, mean quantities (e.g., skin friction), Reynolds stresses, turbulent kinetic energy, and flow visualization. The numerical test case is an approximate computational analog of one of the few stability experiments performed for hypersonic boundary-layer flows. Comparisons and contrasts are drawn between the experimental and the computational results. “Rope-like” waves similar to those observed in schlieren images of high-speed transitional flows are also observed in the numerical experiment and are shown to be visual manifestations of second-mode instability waves.

Eulerian Time-Domain Filtering for Spatial Large-Eddy Simulation

An approach to large-eddy simulation is developed whose subgrid-scale model incorporates Eulerian time-domain filtering, in contrast to conventional approaches that exploit spatial filtering. For applications to large-eddy simulation, time-domain filters enjoy certain advantages relative to spatial filters. Among these, they more naturally commute with differentiation operators than do spatial filters, and there is typically wide separation between numerical truncation error and the dissipation of the subgrid-scale model. Eulerian time-domain filtering is most appropriate for flows whose large coherent structures convect approximately at a common characteristic velocity. Such flows include jets, mixing layers, and wakes. The method is demonstrated in the large-eddy simulation of a heated, subsonic, axisymmetric jet, and results are compared with those obtained from well-resolved direct numerical simulation

On Taylor-series expansions of residual stress

Although subgrid-scale models of similarity type are insufficiently dissipative for practical applications to large-eddy simulation, in recently published a priori analyses, they perform remarkably well in the sense of correlating highly against exact residual stresses. Here, Taylor-series expansions of residual stress are exploited to explain the observed behavior and “success” of similarity models. Specifically, the first few terms of the exact residual stress τkl are obtained in (general) terms of the Taylor coefficients of the grid filter. Also, by expansion of the test filter, a similar expression results for the resolved turbulent stress tensor Lkl in terms of the Taylor coefficients of both the grid and test filters. Comparison of the expansions for τkl and Lkl yields the grid- and test-filter dependent value of the constant cL in the scale-similarity model of Liu et al. [J. Fluid Mech. 275, 83 (1994)]. Until recently, little attention has been given to issues related to the convergence of such exp...

A priori analyses of three subgrid-scale models for one-parameter families of filters

The decay of isotropic turbulence in a compressible flow is examined by direct numerical simulation (DNS). A priori analyses of the DNS data are then performed to evaluate three subgrid-scale (SGS) models for large-eddy simulation (LES): an eddy-diffusivity model (M1) [J. Fluid Mech. 238, 1 (1992)], a stress-similarity model (M2) [J. Fluid Mech. 275, 83 (1994)], and a gradient model (M3) [Theor. Comput. Fluid Dyn. 8, 309 (1996)]. The models exploit one-parameter second- or fourth-order filters of the Pade type, which permit the cutoff wave number kc to be tuned independently of the grid increment Δx. The modeled (M) and exact (E) SGS-stresses are compared component-wise by correlation coefficients of the form C(E,M) computed over the entire three-dimensional fields. In general, M1 correlates poorly against exact stresses (C<0.2), M3 correlates moderately well (C≈0.6), and M2 correlates remarkably well (0.8

An adaptive N -body algorithm of optimal order

Picard iteration is normally considered a theoretical tool whose primary utility is to establish the existence and uniqueness of solutions to first-order systems of ordinary differential equations (ODEs). However, in 1996, Parker and Sochacki [Neural, Parallel, Sci. Comput. 4 (1996)] published a practical numerical method for a certain class of ODEs, based upon modified Picard iteration, that generates the Maclaurin series of the solution to arbitrarily high order. The applicable class of ODEs consists of first-order, autonomous systems whose right-hand side functions (generators) are projectively polynomial; that is, they can be written as polynomials in the unknowns. The class is wider than might be expected. The method is ideally suited to the classical N-body problem, which is projectively polynomial. Here, we recast the N-body problem in polynomial form and develop a Picard-based algorithm for its solution. The algorithm is highly accurate, parameter-free, and simultaneously adaptive in time and order. Test cases for both benign and chaotic N-body systems reveal that optimal order is dynamic. That is, in addition to dependency upon N and the desired accuracy, optimal order depends upon the configuration of the bodies at any instant.

On the nonlinear stability of a high‐speed, axisymmetric boundary layer

The stability of a high‐speed, axisymmetric boundary‐layer flow is investigated by means of secondary instability theory and direct numerical simulation. Parametric studies based on temporal secondary instability theory identify subharmonic secondary instability as a likely path to transition on a hollow cylinder at Mach 4.5. The theoretical predictions are validated by direct numerical solution of the compressible Navier–Stokes equations. Initial perturbations for the temporal direct numerical simulation consist of an axisymmetric ‘‘second‐mode’’ primary disturbance and a subharmonic secondary disturbance comprised of four oblique wave components. At small initial amplitudes of the secondary disturbance, growth rates obtained from the spectrally accurate numerical simulation agree to several significant digits with linear growth rates predicted by secondary instability theory. Qualitative agreement persists to relatively large amplitudes of the secondary disturbance. Moderate transverse curvature is show...

Simulation of crossflow instability on a supersonic highly swept wing

Abstract A direct numerical simulation (DNS) algorithm has been developed for use in the investigation of crossflow instability on supersonic swept wings, an application of potential relevance to the design of the High-Speed Civil Transport (HSCT). The fully explicit algorithm exploits high-order compact-difference and spectral-collocation methods to solve the compressible Navier–Stokes equations in body-fitted coordinates. The method is applied to the investigation of stationary crossflow instability on an infinitely long 77-degree swept wing in Mach 3.5 flow. The results of the DNS are compared with the predictions of linear stability theory (LST) and linear parabolized stability equation (PSE) methodology. In general, the independently conducted DNS and PSE investigations agree closely in terms of the growth rate, the structure, and the orientation angle of the predicted stationary crossflow instability. Although further study is warranted for the case of large-amplitude (nonlinear) disturbances, the close agreement between the methods offers preliminary validation of both the DNS and PSE approaches for this application.

Direct numerical simulation and data analysis of a Mach 4.5 transitional boundary‐layer flow

This paper describes the creation by temporal direct numerical simulation and the analysis based on the Reynolds stress transport equations of a high quality data set that represents the laminar–turbulent transition of a high‐speed boundary‐layer flow. Following Pruett and Zang [Theoret. Comput. Fluid Dyn. 3, 345 (1992)], and with the help of algorithmic refinements, the evolution of an axial, Mach 4.5 boundary‐layer flow along the exterior of a hollow cylinder is simulated numerically. From a perturbed laminar initial state, the well‐resolved simulation proceeds through laminar breakdown to the beginning of a turbulent flow regime. Favre‐averaged Reynolds stress transport equations are derived in generalized curvilinear coordinates and are then specialized to the cylindrical geometry at hand. Reynolds stresses and various turbulence quantities, such as turbulent kinetic energy and turbulent Mach number, are calculated from the numerical data at various stages of the transition process. The kinetic energy...

On the Non-Uniqueness of the Parallel-Flow Approximation

The assumptions and approximations of classical linear stability theory are re-examined to unravel an apparent paradox: the conservative and nonconservative formulations of the linearized disturbance equations can lead to significantly different results when subject to the parallel-flow approximation. We examine closely one such pathological case: boundary-layer flow along a sharp cone.