M. Yousuff Hussaini

On the large‐eddy simulation of transitional wall‐bounded flows

The structure of the subgrid‐scale fields in plane channel flow has been studied at various stages of the transition process to turbulence. The residual stress and subgrid‐scale dissipation calculated using velocity fields generated by direct numerical simulations of the Navier–Stokes equations are significantly different from their counterparts in turbulent flows. The subgrid scale dissipation changes sign over extended areas of the channel, indicating energy flow from the small scales to the large scales. This reversed energy cascade becomes less pronounced at the later stages of transition. Standard residual stress models of the Smagorinsky type are excessively dissipative. Rescaling the model constant improves the prediction of the total (integrated) subgrid scale dissipation, but not that of the local one. Despite the somewhat excessive dissipation of the rescaled Smagorinsky model, the results of a large‐eddy simulation of transition on a flat‐plate boundary layer compare quite well with those of a ...

On spectral multigrid methods for the time-dependent Navier-Stokes equations

A new splitting scheme is proposed for the numerical solution of the time-dependent incompressible Navier-Stokes equations by spectral methods. A staggered grid is used for the pressure, improved intermediate boundary conditions are employed in the split step for the velocity, and spectral multigrid techniques are used for the solution of the implicit equations.

On the linear stability of compressible plane Couette flow

The linear stability of compressible plane Couette flow is investigated. The appropriate basic velocity and temperature distributions are perturbed by a small-amplitude normal-mode disturbance. The full small-amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in some detail. It is found that instabilities can occur, although the corresponding growth rates are often quite small; the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wave speed of the disturbances approaches the velocity of either of the walls, and these regimes are also analysed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

Effects of shock on the stability of hypersonic boundary layers

A set of linearized shock boundary conditions is derived, which is then imposed at the shock to account for the interaction of the shock wave with the boundary/shock layer instability wave; these boundary conditions are used to study the effect of shock on hypersonic boundary layer stability under the assumption of quasi-parallel flow. The result show that the shock has little effect on the boundary layer instability (subsonic first and second mode disturbances) when the shock is located outside the boundary layer edge. When the shock is located near the boundary layer edge, it exerts a stabilizing influence on the first and second modes. The shock also induces unstable supersonic modes with oscillatory structure in the shock layer, but these modes grow slower than the subsonic modes.

Shock-fitted Euler solutions to shock vortex interactions

The interaction of a planar shock wave with one or more vortexes is computed using a pseudospectral method and a finite difference method. The development of the spectral method is emphasized. In both methods the shock wave is fitted as a boundary of the computational domain. The results show good agreement between both computational methods. The spectral method is, however, restricted to smaller time steps and requires use of filtering techniques. Previously announced in STAR as N82-28061

Fourier-Legendre spectral methods for incompressible channel flow

An iterative collocation technique is described for modeling implicit viscosity in three-dimensional incompressible wall bounded shear flow. The viscosity can vary temporally and in the vertical direction. Channel flow is modeled with a Fourier-Legendre approximation and the mean streamwise advection is treated implicitly. Explicit terms are handled with an Adams-Bashforth method to increase the allowable time-step for calculation of the implicit terms. The algorithm is applied to low amplitude unstable waves in a plane Poiseuille flow at an Re of 7500. Comparisons are made between results using the Legendre method and with Chebyshev polynomials. Comparable accuracy is obtained for the perturbation kinetic energy predicted using both discretizations.

Theory of Stability and Convergence for Spectral Methods

In this chapter we present a fairly general approach to the stability and convergence analysis of spectral methods. We confine ourselves to linear problems. Analysis of several non-linear problems is presented in Chaps. 11 and 12. For time-dependent problems, only the discretizations in space are considered. Stability for fully discretized time-dependent problems is discussed in Chap. 4 by a classical eigenvalue analysis, and in Chap. 12 by variational methods.

Fundamentals of Spectral Methods for PDEs

For the remainder of this book we shall be concerned with the use of spectral methods to approximate solutions to partial differential equations (PDEs). Our concern in this chapter is to illustrate how spectral methods are actually implemented for PDEs. We start by deriving the semi-discrete (continuous in time) ODE equations which are satisfied by various spectral approximations to Burgers equation. This will involve a discussion of non-linear terms, boundary conditions, projection operators, and different spectral discretizations. The second section provides a detailed discussion of transform methods for evaluating convolution sums. Next, we discuss Neumann, Robin and radiation boundary conditions. Finally, we remark on the treatment of coordinate singularities and the use of mapping techniques in two-dimensional problems.

Some Algorithms for Unsteady Navier—Stokes Equations

There are essentially four different formulations of the incompressible Navier—Stokes equations—the primitive-variable (velocity and pressure), streamfunction-vorticity, streamfunction and velocity-vorticity formulations. The primitive-variable formulation can be found in any text on fluid dynamics (e.g., Batchelor (1967)), as can the two-dimensional version of both stream-function formulations. The three-dimensional streamfunction equations are given by Murdock (1986). Both streamfunction formulations avoid the complications of dealing with the pressure, as does the straightforward velocityvorticity approach. Murdock (1977, 1986) and Vanel, Peyret and Bontoux (1985) have developed spectral algorithms based on the two-dimensional streamfunction-vorticity formulation. Murdock (1986) has extended this to three dimensions. Gottlieb and Orszag (1977) and Maday and Metivet (1986) discuss spectral methods for the pure streamfunction version in two-dimensional flows. The velocity-vorticity formulation (Dennis, Ingham and Cook (1979)) has not yet been employed in spectral calculations. The primitive-variable formulation has been the one most extensively employed in three-dimensional spectral calculations and spectral methods based on it will be the focus of this chapter.

Solution Techniques for Implicit Spectral Equations

The solution of implicit equations is an important component of many spectral algorithms. For steady problems this task is unavoidable, while spectral algorithms for many unsteady problems are only feasible if they incorporate implicit (or semi-implicit) time discretizations (see Sec. 3.1 and Chap. 7). We concentrate on linear systems, assuming that non-linear ones are attacked by standard linearization techniques.