Spectral methods in fluid dynamics

Fundamental aspects of spectral methods are introduced. Recent developments in spectral methods are reviewed with an emphasis on collocation techniques. Their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed. The key role that these methods play in the simulation of stability, transition, and turbulence is brought out. A perspective is provided on some of the obstacles that prohibit a wider use of these methods, and how these obstacles are being overcome

Multiple paths to subharmonic laminar breakdown in a boundary layer

Numerical simulations demonstrate that laminar breakdown in a boundary layer induced by the secondary instability of two-dimensional Tollmien-Schlichting waves to three-dimensional subharmonic disturbances need not take the conventional lambda vortex/high-shear layer path

Numerical experiments on the stability of controlled boundary layers

Nonlinear simulations are presented for instability and transition in parallel water boundary layers subjected to pressure gradient, suction, or heating control. In the nonlinear regime, finite amplitude, 2-D Tollmein-Schlichting waves grow faster than is predicted by linear theory. Moreover, this discrepancy is greatest in the case of heating control. Likewise, heating control is found to be the least effective in delaying secondary instabilities of both the fundamental and subharmonic type. Flow field details (including temperature profiles) are presented for both the uncontrolled boundary layer and the heated boundary layer

Iterative spectral methods and spectral solutions to compressible flows

A spectral multigrid scheme is described which can solve pseudospectral discretizations of self-adjoint elliptic problems in O(N log N) operations. An iterative technique for efficiently implementing semi-implicit time-stepping for pseudospectral discretizations of Navier-Stokes equations is discussed. This approach can handle variable coefficient terms in an effective manner. Pseudospectral solutions of compressible flow problems are presented. These include one dimensional problems and two dimensional Euler solutions. Results are given both for shock-capturing approaches and for shock-fitting ones

A three-dimensional spectral algorithm for simulations of transition and turbulence

A spectral algorithm for simulating three dimensional, incompressible, parallel shear flows is described. It applies to the channel, to the parallel boundary layer, and to other shear flows with one wall bounded and two periodic directions. Representative applications to the channel and to the heated boundary layer are presented

Vermiculture bio-technology: An effective tool for economic and environmental sustainability

Vermicompost production and use is an ‘environment friendly, protective and restorative’ process as it diverts waste from ending up in landfills and also reduces emission of greenhouse gases (GHG) due to very small amount of energy used in its production process. Application of vermicompost in farm soil acts as soil conditioner and help to improve its physical, biological and chemical properties. Vermicompost production is also an ‘economically productive’ process as it ‘reduces wastes’ at source and consequently saves landfills space. Construction of engineered landfills incurs 20 to 25 million US dollars upfront before the first load of waste is dumped. Over the past five years, the cost of landfill disposal of waste increased from $29 to$65 per ton of waste in Australia. However, landfills have to be monitored for at least 30 years for emissions of GHG and toxic gases and leachate (Waste Juice) which also incur cost. During 2002 to 2003, waste management services within Australia costed $2458.2 million. Even in developing nations where there are no true landfills, dumping of wastes cost alot on government. This paper elaborates on the importance of this technology to environmental sustainability and economic empowerment.Key words: Vermicompost, earthworm, nitrogen (N), phosphorus (P), potassium (K) and magnesium (Mg)

Psuedospectral calculation of shock turbulence interactions

A Chebyshev-Fourier discretization with shock fitting is used to solve the unsteady Euler equations. The method is applied to shock interactions with plane waves and with a simple model of homogeneous isotropic turbulence. The plane wave solutions are compared to linear theory

Spectral multigrid methods for the solution of homogeneous turbulence problems

New three-dimensional spectral multigrid algorithms are analyzed and implemented to solve the variable coefficient Helmholtz equation. Periodicity is assumed in all three directions which leads to a Fourier collocation representation. Convergence rates are theoretically predicted and confirmed through numerical tests. Residual averaging results in a spectral radius of 0.2 for the variable coefficient Poisson equation. In general, non-stationary Richardson must be used for the Helmholtz equation. The algorithms developed are applied to the large-eddy simulation of incompressible isotropic turbulence

Spectral collocation methods

This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2

On similarity solutions of a boundary layer problem with an upstream moving wall

The problem of a boundary layer on a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream is examined. It was previously shown that the solution of this boundary value problem is crucially dependent on the parameter which is the ratio of the velocity of the plate to the velocity of the free stream. In particular, it was proved that a solution exists only if this parameter does not exceed a certain critical value, and numerical evidence was adduced to show that this solution is nonunique. Using Crocco formulation the present work proves this nonuniqueness. Also considered are the analyticity of solutions and the derivation of upper bounds on the critical value of wall velocity parameter