Philippe R. Spalart

Direct simulation of a turbulent boundary layer up to R θ = 1410

The turbulent boundary layer on a flat plate, with zero pressure gradient, is simulated numerically at four stations between Rθ = 225 and Rθ = 1410. The three-dimensional time-dependent Navier-Stokes equations are solved using a spectral method with up to about 107 grid points. Periodic spanwise and streamwise conditions are applied, and a multiple-scale procedure is applied to approximate the slow streamwise growth of the boundary layer. The flow is studied, primarily, from a statistical point of view. The solutions are compared with experimental results. The scaling of the mean and turbulent quantities with Reynolds number is compared with accepted laws, and the significant deviations are documented. The turbulence at the highest Reynolds number is studied in detail. The spectra are compared with various theoretical models. Reynolds-stress budget data are provided for turbulence-model testing.

Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions

Abstract Two numerical methods were designed to solve the time-dependent, three-dimensional, incompressible Navier-Stokes equations in boundary layers (method A, semi-infinite domain) and mixing layers or wakes (method B, fully-infinite domain). Their originality lies in the use of rapidly-decaying spectral basis functions to approximate the vertical dependence of the solutions, combined with one (method A) or two (method B) slowly-decaying “extra functions” for each wave-vector that exactly represent the irrotational component of the solution at large distances. Both methods eliminate the pressure term as part of the formulation, thus avoiding fractional-step time integration. They yield rapid convergence and are free of spurious modes in the Orr-Sommerfeld spectra. They are also efficient, although the operation count is of order N 2 ( N is the number of modes in the infinite direction). These methods have been used for extensive direct numerical simulations of transition and turbulence. A new time-integration scheme, with low storage requirements and good stability properties, is also described.

Linear and nonlinear stability of the Blasius boundary layer

Two new techniques for the study of the linear and nonlinear instability in growing boundary layers are presented. The first technique employs partial differential equations of parabolic type exploiting the slow change of the mean flow, disturbance velocity profiles, wavelengths, and growth rates in the streamwise direction. The second technique solves the Navier–Stokes equation for spatially evolving disturbances using buffer zones adjacent to the inflow and outflow boundaries. Results of both techniques are in excellent agreement. The linear and nonlinear development of Tollmien–Schlichting (TS) waves in the Blasius boundary layer is investigated with both techniques and with a local procedure based on a system of ordinary differential equations. The results are compared with previous work and the effects of non-parallelism and nonlinearly are clarified. The effect of nonparallelism is confirmed to be weak and, consequently, not responsible for the discrepancies between measurements and theoretical results for parallel flow. Experimental uncertainties, the adopted definition of the growth rate, and the transient initial evolution of the TS wave in vibrating-ribbon experiments probably cause the discrepancies. The effect of nonlinearity is consistent with previous weakly nonlinear theories. White nonlinear effects are small near branch I of the neutral curve, they are significant near branch II and delay or event prevent the decay of the wave.

A numerical study of the turbulent Ekman layer

The three-dimensional time-dependent turbulent flow in a neutrally stratified Ekman layer over a smooth surface is computed numerically by directly solving the Navier–Stokes equations. All the relevant scales of motion are included in the simulation so that no turbulence model is needed. Results of the simulations indicate that the horizontal component of the rotation vector has a significant influence on the turbulence; thus the ‘f-plane’ approximation fails. Differences as large as 20% in the geostrophic drag coefficient, u*/G, and 70% in the angle between the freestream velocity and the surface shear stress are found, depending on the latitude and the direction of the geostrophic wind. At 45° latitude, differences of 6 and 30% are noted in the drag coefficient and the shear angle, respectively, owing to the variation of the wind direction alone. Asymptotic similarity theory and a higher-order correction are first tested for the range of low Reynolds numbers simulated, and then used to predict the friction velocity and stress direction at the surface for flows at arbitrary Reynolds number. A model for the variation of these quantities with latitude and wind angle is also proposed which gives an acceptable fit to the simulation results. No large-scale longitudinal vortices are found in the velocity fields, reinforcing the conjecture that unstable thermal stratification, in addition to inflectional instability, is required to produce and maintain the large-scale rolls observed in the Earths boundary layer. Comparisons of the Ekman layer with a related three-dimensional boundary layer reveal similarities of the mean profiles, as well as qualitative differences.

Numerical study of sink-flow boundary layers

Direct numerical simulations of sink-flow boundary layers, with acceleration parameters K between 1.5 x 10 to the -6th and 3.0 x 10 to the -6th, are presented. The three-dimensional, time-dependent Navier-Stokes equations are solved numerically, using a spectral method, with about one million degrees of freedom. The flow is assumed to be statistically steady, and self-similar. A multiple-scale approximation and periodic conditions are applied to the fluctuations. The turbulence is studied using instantaneous and statistical results. Good agreement with the experiments of Jones and Launder (1972) is observed. The two effects of the favorable pressure gradient are to extend the logarithmic layer, and to alter the energy balance of the turbulence near the edge of the boundary layer. At low Reynolds number the logarithmic layer is shortened and slightly displaced, but wall-layer streaks are present even at the lowest values of R(theta) for which turbulence can be sustained. Large quiescent patches appear in the flow. Relaminarization occurs at K = 3.0 x 10 to the -6th, corresponding to a Reynolds number R(theta) of about 330.

Numerical study of ribbon-induced transition in Blasius flow

The early three-dimensional stages of transition in the Blasius boundary layer are studied by numerical solution of the Navier-Stokes equations. A finite-amplitude two-dimensional wave and low-amplitude three-dimensional random disturbances are introduced. Rapid amplification of the three-dimensional components is observed and leads to transition. For intermediate amplitudes of the two-dimensional wave the breakdown is of subharmonic type, and the dominant spanwise wavenumber increases with the amplitude. For high amplitudes the energy of the fundamental mode is comparable to the energy of the subharmonic mode, but never dominates it; the breakdown is of mixed type. Visualizations, energy histories, and spectra are presented. The sensitivity of the results to various physical and numerical parameters is studied. The agreement with experimental and theoretical results is discussed.

Direct Simulation of a Turbulent Oscillating Boundary Layer

The turbulent boundary layer under a freestream velocity that varies sinusoidally in time around a zero mean is considered. The flow has a rich variety of behaviors including strong pressure gradients, inflection points in the velocity profile, and reversal of the shear stress. A theory for the velocity- and stress profiles at high Reynolds number is formulated. Well-resolved direct Navier-Stokes simulations are conducted over a narrow range of Reynolds numbers. The flow is also computed over a wider range of Reynolds numbers using a new algebraic turbulence model. The results produced by the three approaches and by experiments are compared. Detailed phase-averaged statistical results from the direct simulations are provided to assist turbulence-model development.

Numerical simulations of turbulent spots in plane Poiseuille and boundary‐layer flow

Direct numerical simulations of turbulent spots in plane Poiseuille and boundary‐layer flows are performed. Mature, self‐similar spots are obtained. The propagation velocities and spreading angles are found to compare well with corresponding experiments. The difference in shape of the two spots is also clearly discernible: the turbulent parts are contained within arrowhead regions that point in opposite directions for the two cases. The wing‐tip region of the Poiseuille spot is also found to consist of a large‐amplitude semiturbulent wave packet.

Direct simulation of the stably stratified turbulent Ekman layer

The three-dimensional time-dependent turbulent flow in the stably stratified Ekman layer over a smooth surface is computed numerically by directly solving the Navier–Stokes equations, using the Boussinesq approximation to account for buoyancy effects. All relevant scales of motion are included in the simulation so that no turbulence model is needed. The Ekman layer is an idealization of the Earths boundary layer and provides information concerning atmospheric turbulence models. We find that, when non-dimensionalized according to Nieuwstadts local scaling scheme, some of the simulation data agree very well with atmospheric measurements. The results also suggest that Brost & Wyngaards ‘constant Froude number’ and Hunts ‘shearing length’ stable layer models for the dissipation rate of turbulent kinetic energy are both valid, when Reynolds number effects are accounted for. Simple gradient closures for the temperature variance and heat flux demonstrate the same variation with Richardson number as in Mason & Derbyshires large-eddy simulation (LES) study, implying both that the models are relatively insensitive to Reynolds number and that local scaling should work well when applied to the stable atmospheric layer. In general we find good agreement between the direct numerical simulation (DNS) results reported here and Mason & Derbyshires LES results.

Theoretical and numerical study of a three-dimensional turbulent boundary layer

The boundary layer is created on an infinite flat plate by a time-dependent free-stream velocity vector, whose magnitude is independent of time but whose direction (as seen in plan view) changes at a constant angular velocity. The pressure gradient, at right angles to the free-stream velocity, induces a skewing of the velocity profile; all components of the Reynolds-stress tensor are non-zero (using axes aligned with the wall and the flow direction). This flow has never been produced experimentally, but it has the merit of being simply defined and of having only the Reynolds number as a parameter, which greatly simplifies the analysis. The flow is studied theoretically using Reynolds-number scaling laws, and by direct numerical simulation over a range of Reynolds numbers. The simplest version of the theory is equivalent to existing theories of the Ekman layer. A higher-order version is presented and yields excellent agreement with the numerical results at three Reynolds numbers, with just one adjustable constant in each equation. The theory allows the extrapolation of the results to high Reynolds numbers. The Reynolds-averaged equations reduce to a one-dimensional steady problem, so that turbulence-model testing will be easy and accurate. Detailed data are provided for that purpose.