Singularity Formation in the Geometry of Perturbed Shocks of General Mach Number

While planar shock waves are known to be stable to small perturbations in the sense that the perturbation amplitude decays over time, it has also been suggested that plane propagating shocks can develop singularities in some derivative of their geometry (Whitham (1974) Linear and nonlinear waves. Wiley, New York) in a nonlinear, wave reinforcement process. We present a spectral-based analysis of the equations of geometrical shock dynamics that predicts the time to singularity formation in the profile of an initially perturbed planar shock for general shock Mach number. We find that following an initially sinusoidal perturbation, the shock shape remains analytic only up to a finite, critical time that is a monotonically decreasing function of the initial perturbation amplitude. At the critical time, the shock profile ceases to be analytic, corresponding physically to the incipient formation of a “shock-shock” or triple point. We present results for gas-dynamic shocks and discuss the potential for extension to shock dynamics of fast MHD shocks

Hypervelocity cone-flow with reaction chemistry by second order kinetic theory based Euler solver

Calculations are presented of the hypervelocity Euler flow of nitrogen past a cone at incidence, at an enthalpy sufficiently high to produce dissociation/recombination chemistry downstream o the bow shock wave. A spatially second order accurate scheme for the numerical solution of the Euler equations based on the kinetic theory of gases is used, combined with the Lighthill-Freeman model of the non-equilibrium ideal dissociating gas. The result depict the effect of the interaction between gas dynamics and finite rate gas chemistry on the shape and strength of the shock induced vortex system near the flowfield leeward plane of symmetry

A Computational Investigation of Inviscid Hypervelocity Flow of a Dissociating Gas Past a Cone at Incidence

Calculations have been performed for the inviscid hypervelocity flow of nitrogen past a 15 degree semi-angle sharp cone at an incidence of 30 degree, at an enthalphy sufficiently high to produuce dissociation/recombination chemistry downstream of the bow shock wave. A spatially second-order-accurate EFM (Equilibrium Flux Method) scheme for the numerical solution of the inviscid Euler equations was used, combined with the Lighthill-Freeman model of the non-equilibrium ideal dissociating gas. The computations have been sued to gain an understanding fo the interaction between the ags dynamics and the finite-rate chemistry. Inviscid flow has been considered to ensure that the only physical length scales in the flow are those associated with the chemical reactions. It was found that a chemical length scale L_s, based on the local dissociation length behind the shock on the windward plane of symmetry is an important governing parameter of the flow. However, as the flow length-scale becomes large and the flow approached the limiting case of equilibrium chemistry, L_s is not the dominant chemical length-scale, particularly in the leeward flow which contains a shock-vortex structure. A simple modelling technique has been used to determine a more appropriate length scale L_r for the leeward flow, based on the equilibrium conditions behind the leeward cross-flow shock

Large eddy simulation of smooth-wall, transitional and fully rough-wall channel flow

Large eddy simulation (LES) is reported for both smooth and rough-wall channel flows at resolutions for which the roughness is subgrid. The stretched vortex, subgrid-scale model is combined with an existing wall-model that calculates the local friction velocity dynamically while providing a Dirichlet-like slip velocity at a slightly raised wall. This wall model is presently extended to include the effects of subgrid wall roughness by the incorporation of the Hama's roughness function ΔU^+(k^+_(s∞)) that depends on some geometric roughness height k_(s∞) scaled in inner variables. Presently Colebrook's empirical roughness function is used but the model can utilize any given function of an arbitrary number of inner-scaled, roughness length parameters. This approach requires no change to the interior LES and can handle both smooth and rough walls. The LES is applied to fully turbulent, smooth, and rough-wall channel flow in both the transitional and fully rough regimes. Both roughness and Reynolds number effects are captured for Reynolds numbers Re_b based on the bulk flow speed in the range 10^4–10^(10) with the equivalent Re_τ, based on the wall-drag velocity u_τ varying from 650 to 10^8. Results include a Moody-like diagram for the friction factor f = f(Re_b, ∈), ∈ = k_(s∞)/δ, mean velocity profiles, and turbulence statistics. In the fully rough regime, at sufficiently large Re_b, the mean velocity profiles show collapse in outer variables onto a roughness modified, universal, velocity-deficit profile. Outer-flow stream-wise turbulence intensities scale well with u_τ for both smooth and rough-wall flow, showing a log-like profile. The infinite Reynolds number limits of both smooth and rough-wall flows are explored. An assumption that, for smooth-wall flow, the turbulence intensities scaled on u_τ are bounded above by the sum of a logarithmic profile plus a finite function across the whole channel suggests that the infinite Re_b limit is inviscid slip flow without turbulence. The asymptote, however, is extremely slow. Turbulent rough-wall flow that conforms to the Hama model shows a finite limit containing turbulence intensities that scale on the friction factor for any small but finite roughness

Calculations of three-dimensional hypervelocity cone-flow with chemical reactions

We present the results of CFD calculations of the hypervelocity flow of nitrogen about a slightly blunted cone at an angle of incidence. We use Pullin's Equilibrium Flux Method, and include the effect of finite rate dissociation and recombination chemical reactions vis the Lighthill-Freeman model of the ideal dissociating gas. Test results with the chemical reactions frozen are compared with the perfect gas calculations of Marconi for the same flow geometry and Mach number. The results for chemical reacting flow show substantial interaction between the gas dynamics and the chemistry on the leeward cone surface

A Numerical Study of Hypersonic Leeward Flow Over a Delta Wing Using a Parallel Architecture Supercomputer

We have implemented Pullin's kinetic theory-based Equilibrium Flux Method [Pullin, J. Comput. Physics, 34:231-244, 1980] on a parallel supercomputer (the Intel iPSC Touchstonde Delta). We study the leeward flowfield of a blunt-nosed delta-wing at an angle of attack of 30 degrees and freestream Mach number of 8.7. The delta-wing sweep angle is 70 degrees. The freestream Reynolds number varies from 100,000 to 1,000,100 and the wing temperature is fixed at 300 K. Computational results are presented for a series of grids for both inviscid and laminar viscous flows. Of particular interest are the vortex and shock structures in the leeward flow that are evident only with the high grid resolution. Some experiences of developing a CFD code for a supercomputer are discussed

A Niche-Based Framework to Assess Current Monitoring of European Forest Birds and Guide Indicator Species' Selection

Concern that European forest biodiversity is depleted and declining has provoked widespread efforts to improve management practices. To gauge the success of these actions, appropriate monitoring of forest ecosystems is paramount. Multi-species indicators are frequently used to assess the state of biodiversity and its response to implemented management, but generally applicable and objective methodologies for species' selection are lacking. Here we use a niche-based approach, underpinned by coarse quantification of species' resource use, to objectively select species for inclusion in a pan-European forest bird indicator. We identify both the minimum number of species required to deliver full resource coverage and the most sensitive species' combination, and explore the trade-off between two key characteristics, sensitivity and redundancy, associated with indicators comprising different numbers of species. We compare our indicator to an existing forest bird indicator selected on the basis of expert opinion and show it is more representative of the wider community. We also present alternative indicators for regional and forest type specific monitoring and show that species' choice can have a significant impact on the indicator and consequent projections about the state of the biodiversity it represents. Furthermore, by comparing indicator sets drawn from currently monitored species and the full forest bird community, we identify gaps in the coverage of the current monitoring scheme. We believe that adopting this niche-based framework for species' selection supports the objective development of multi-species indicators and that it has good potential to be extended to a range of habitats and taxa