Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Theoretical and Computational Approach to Modeling Flame Ignition.

Abstract: : In this paper, time-dependent results obtained from a simplified but nonlinear analytic similarity solution and a detailed numerical simulation are used to study the relations between the fundamental processes occurring in the very early stages of flame ignition in homogeneous premixed gases. The parameters which may be varied are the composition of the mixture, the initial radius of energy deposition R sub 0, the duration of the heating TAN SUB 0, and the total energy deposited in the system E sub 0. The similarity solution plus the ignition delay times Tau sub c for the fuel-oxidizer mixture as a function of temperature can be used to calculate whether or not a given energy source is adequate to ignite the system. This simple procedure may then be calibrated using the time-dependent NRL detailed reactive flow models. The models include the thermophysical properties of the mixture, a full chemical kinetics scheme, the nonlinear convection of self-consistent fluid dynamics and the matrix molecular diffusion coefficients for the individual species. Results are presented for a selected mixture of H2-O2-N2 for two values of R sub 0 which show that a model must be constructed for a quench volume in order to complete the similarity solution calibration. (Author)

The propagation and basal solidification of two-dimensional and axisymmetric viscous gravity currents

Abstract: The effect of basal solidification on viscous gravity currents is analysed using continuum models. A Stefan condition for basal solidification is incorporated into the Navier-Stokes equations. A simplified version of this model is determined in the lubrication and large-Bond-number limit. Asymptotic solutions are obtained in three parameter regimes. (i) A similarity solution is possible in the following cases: the two-dimensional problem when volume per unit length (V) is proportional to time (t) raised to the power 7/4(V = qt7/4) and the Julian number (v3g2 /q4 ) is large, where v is kinematic viscosity, q is a constant of proportionality and g is the acceleration due to gravity; the axisymmetric problem when volume is proportional to time raised to the power 3 (V = Qt3) and the dimensionless group vg/Q is large, where Q is a constant of proportionality. In both cases, the front is found to depend on time raised to the power 5/4, as it does in the absence of solidification, but the constant of proportionality satisfies a modified system of equations. (ii) In the case of large Stefan number and small modified Peclet number (Peδ2 ≪ 1, where Pe is the Peclet number and δ is the aspect ratio), asymptotic and numerical methods are combined to produce the most revealing results. The temperature of the fluid approaches the melting point over a short time-scale. Over the long time-scale, the solid/liquid interface is determined from the conduction of latent heat into the solid. Strong coupling is observed with the basal solidification modifying the flow at leading order. The solidification may retard and eventually arrest the front motion long before complete phase change has taken place. (iii) In the case of constant volume and large modified Peclet number (Peδ2 ≫ 1), similarity solutions are found for the solidification at the base of the gravity current on the short time-scale. The coupling is weak on this time-scale with the solidification being dependent on the front position but not influencing the fluid motion at leading order. Over the long time-scale, the drop completely solidifies. Analytical solutions are not obtained on this time-scale, but scalings are deduced.

Verification Assessment of Piston Boundary Conditions for Lagrangian Simulation of Compressible Flow Similarity Solutions

Abstract: This work is concerned with the use of similarity solutions of the compressible flow equations as benchmarks or verification test problems for finite-volume compressible flow simulation software. In practice, this effort can be complicated by the infinite spatial/temporal extent of many candidate solutions or “test problems.” Methods can be devised with the intention of ameliorating this inconsistency with the finite nature of computational simulation; the exact strategy will depend on the code and problem archetypes under investigation. For example, self-similar shock wave propagation can be represented in Lagrangian compressible flow simulations as rigid boundary-driven flow, even if no such “piston” is present in the counterpart mathematical similarity solution. The purpose of this work is to investigate in detail the methodology of representing self-similar shock wave propagation as a piston-driven flow in the context of various test problems featuring simple closed-form solutions of infinite spatial/temporal extent. The closed-form solutions allow for the derivation of similarly closed-form piston boundary conditions (BCs) for use in Lagrangian compressible flow solvers. The consequences of utilizing these BCs (as opposed to directly initializing the self-similar solution in a computational spatial grid) are investigated in terms of common code verification analysis metrics (e.g., shock strength/position errors and global convergence rates).