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doi: 10.1002/fld.1650210202pmid: N/A
The problem of finding the shape of a body with smallest drag in a flow governed by the two‐dimensional steady Navier‐Stokes equations is considered. The flow is expressed in terms of a streamfunction which satisfies a fourth‐order partial differential equation with the biharmonic operator as principal part. Using the adjoint variable approach, both the first‐ and second‐order necessary conditions for the shape with smallest drag are obtained. An algorithm for the calculation of the optimal shape is proposed in which the first variations of solutions of the direct and adjoint problems are incorporated. Numerical examples show that the algorithm can produce the optimal shape successfully.
doi: 10.1002/fld.1650210203pmid: N/A
An improved low‐Reynolds‐number k‐ϵ model has been formulated and tested against a range of DNS (direct numerical simulation) and experimental data for channel and complex shear layer flows. The model utilizes a new form of damping function adopted to account for both wall proximity effects and viscosity influences and a more flexible damping argument based on the gradient of the turbulent kinetic energy on the wall. Additionally, the extra production of the inhomogeneous part of the viscous dissipation near a wall has been added to the dissipation equation with significantly improved results. The proposed model was successfully applied to the calculation of a range of wall shear layers in zero, adverse and favourable pressure gradients as well as backward‐facing‐step separated flows.
doi: 10.1002/fld.1650210204pmid: N/A
A new numerical scheme for reacting axisymmetric jet flows formed between a fuel jet and co‐flowing air has been developed. The model is mathematically described by a set of non‐linear parabolic partial differential equations in two space dimensions, i.e. the boundary layer equations. The numerical scheme that the programme uses for solving the fully coupled conservation equations of mass, momentum, energy and species is a generaliztion of the discretization technique recently developed by Villasenor (J. Math. Comput. Simul., 36, 203–208 (1994)). Chemical production (and destruction) of the species is allowed to occur through N elementary reversible (or irreversible) reactions involving k species, although in the present model the reaction rates are evaluated with a simplified kinetic mechanism for a one‐step global reaction. Thermal radiation is considered assuming an optically thin limit and adopting the grey medium approximation. Allowances are made for natural convection effects and variable thermodynamic and molecular transport properties. The performance of the model in solving the coupled aerodynamic and finite rate chemistry effects is tested by comparing model predictions with experimental data of Mitchell et al. (Combust. Flame, 37, 227–244 (1980)) for a buoyant, laminar, diffusion axisymmetric methane‐air flame.
doi: 10.1002/fld.1650210205pmid: N/A
Peyret (J. Fluid Mech., 78, 49–63 (1976)) and others have described artificial compressibility iteration schemes for solving implicit time discretizations of the unsteady incompressible Navier‐Stokes equations. Such schemes solve the implicit equations by introduing derivatives with respect to a pseudo‐time variable τ and marching out to a steady state in τ. The pseudo‐time evolution equation for the pressure p takes the form ∂p/∂ = −a2∂∇.u, where a is an artificial compressibility parameter and u is the fluid velocity vector. We present a new scheme of this type in which convergence is accelerated by a new procedure for setting a and by introducing an artificial bulk viscosity b into the momentum equation. This scheme is used to solve the non‐linear equations resulting from a fully implicit time differencing scheme for unsteady incompressible flow. We find that the best values of a and b are generally quite different from those in the analogous scheme for steady flow (J. D. Ramshaw and V. A. Mousseau, Comput. Fluids, 18, 361–367 (1990)), owing to the previously unrecognized fact that the character of the system is profoundly altered by the pressence of the physical time derivative terms. In particular, a Fourier dispersion analysis shows that a no longer has the significance of a wave speed for finite values of the physical time step δt,. Inded, if on sets a ˜ |u| as usual, the artificial sound waves cease to exist when δt is small and this adversely affects the iteration convergence rate. Approximate analytical expressions for a and b are proposed and the benefits of their use relative to the conventional values a ∼ |u| and b = 0 are illustrated in simple test calculations.
doi: 10.1002/fld.1650210206pmid: N/A
The disarrangement of a perturbed lattice of vortices was studied numerically. The basic state is an exponentially decaying, exact solution of the Navier‐Stokes equations. Square arrays of vortices with even numbers of vortex cells along each side were perturbed and their evolution was investigated. Whether the energy in the perturbation grows somewhat before it decays or decays monotonically depends on the initial strength of the vortices of the basic state, the extent of lateral confinement and the structure of the perturbation. The critical condition for temporally local instability, i.e. the critical amplitude of the basic state that must be exceeded to allow energy transfer from the basic state to the perturbation, is discussed. In the strongly confined case of a square lattice of four vortices the appearance of enchancement of global rotation is the result of energy transfer from the basic state to a temporally local unstable mode. Energy is transferred from the basic state to larger‐scaled structures (inverse cascade) only if the scales of the larger structures are inherently contained in the initial structure of the perturbation. The initial structure of the double array of vortices is not maintained except for a very special form of perturbation. The facts that large scales decay more slowly than small scales and that, when non‐linearities are sufficiently strong, energy is transferred from one scale to another explain the differences in the disarrangement process for different initial strengths of the vortices of the basic state. The stronger vortices, i.e. the vortices perturbed in a manner that increases their strength, tend to dominate the weaker vortices. The pairing and subsequent merging (or capture) of vortices of like sense into larger‐scale vortices are described in terms of peaks in the evolution of the square root of the palinstrophy divided by the enstrophy.
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