Temporal Direct Numerical Simulation Research Articles

Parallel performance assessment of a pseudo-spectral solver for transition and turbulent boundary layer flows

The computational performance and accuracy of a parallel pseudo-spectral solver is assessed, and the solver is validated for direct numerical simulations (DNS) of bypass transition and turbulent boundary layer flows. The solver shows spatial order of accuracy of 6.8 and 3.7 for laminar and turbulent flow simulations, respectively, and scalability up to 16 thousand processors. DNS predictions for plane channel and flat plate bypass transition flows compare very well with the benchmark results available in the literature. Analysis of the turbulent structures shows counter-rotating longitudinal structures in the sub/buffer layers, which inducing lifting on each other to generate series of sweep and ejection events. These events generate shear stress which transfers energy from mean flow to streamwise turbulent fluctuations. The energy is then redistributed to other turbulent velocity fluctuations via pressure-strain. Temporal DNS of bypass pass transition flow compare very well with spatial DNS using an order of magnitude lower computational cost, thus is identified to be a viable alternative for transition flow physics simulations.

Integral invariants and decay of temporally developing grid turbulence

We present a study of a large-scale energy spectrum and integral invariants in temporally developing grid turbulence at mesh Reynolds numbers of ReM = 10 000 and 20 000 by employing direct numerical simulations (DNSs) in a periodic box. The simulations are initialized with a velocity field that approximates the wakes induced by the bars of conventional square grids. The turbulence statistics obtained in the temporal DNS agree well with those of the previous experiments in both the production and decay regions. The temporally developing grid turbulence also has a so-called non-equilibrium region, which is consistent with its spatially developing counterpart, where the normalized dissipation rate of turbulence kinetic energy (TKE), Cε, increases as the turbulence decays. The decay exponent n of TKE is n = 1.22 at ReM = 20 000 and n = 1.35 at ReM = 10 000, which are close to the values for the Saffman turbulence [i.e., 6/5 for ReM = 20 000 and 6(1 + p)/5 ≈ 1.36 for ReM = 10 000 with p ≈ 0.13 obtained by Cε ∼ tp at large t]. The longitudinal integral length scale and the TKE dissipation rate also exhibit temporal evolutions consistent with the Saffman turbulence for both ReM. The Saffman integral directly evaluated in the grid turbulence tends to be time-independent after the turbulence evolves for about 200 times of characteristic time scale defined by mesh size divided by the mean velocity of a fluid passing the grid. A direct evaluation of the TKE spectrum E(k) shows that E(k) ≈ Lk2/4π2 is valid for a finite range of low wavenumbers.

Temporal direct numerical simulation of transitional natural-convection boundary layer under conditions of considerable external turbulence effects

Results of direct numerical simulations for time-developing air natural-convection boundary layer are presented. Computations have been performed assuming periodicity conditions in both the directions parallel to the vertical isothermal hot plate. The contribution is mainly focused on understanding of laminar–turbulent transition peculiarities in the case of perturbation action of external turbulence that is modeled by isotropic disturbances initially introduced into the computational domain. Special attention is paid to identification and analysis of evolving three-dimensional vortices that clearly manifest themselves through the whole stages of laminar–turbulent transition in the boundary layer. A comparison of computed profiles of mean velocity, mean temperature and fluctuation characteristics for turbulent regimes of convection with experimental data is performed as well.

Extending temporal simulations

Direct numerical simulations (DNS) of spatially growing turbulent shear layers may be performed as temporal simulations by solving the governing equations with some additional terms while imposing streamwise periodicity. These terms are functions of the means whose spatial growth is calculated easily and accurately from statistics of the temporal DNS. Equations for such simulations are derived.

Inflow conditions for spatial direct numerical simulation of turbulent boundary layers

The inflow conditions for spatial direct numerical simulation (SDNS) of turbulent boundary layers should reflect the characteristics of upstream turbulence, which is a puzzle. In this paper a new method is suggested, in which the flow field obtained by using temporal direct numerical simulation (TDNS) for fully developed turbulent flow (only flow field for a single moment is sufficient) can be used as the inflow of SDNS with a proper transformation. The calculation results confirm that this method is feasible and effective. It is also found that, under a proper time-space transformation, all statistics of the fully developed turbulence obtained by both temporal mode and spatial mode DNS are in excellent agreement with each other, not only qualitatively, but also quantitatively. The normal-wise distributions of mean flow profile, turbulent Mach number and the root mean square (RMS) of the fluctuations of various variables, as well as the Reynolds stresses of the fully developed turbulence obtained by using SDNS, bear similarity in nature.

Transition thresholds in the asymptotic suction boundary layer

Energy thresholds for transition to turbulence in an asymptotic suction boundary layer is calculated by means of temporal direct numerical simulations. The temporal assumption limits the analysis to periodic disturbances with horizontal wave numbers determined by the computational box size. Three well known transition scenarios are investigated: oblique transition, the growth and breakdown of streaks triggered by streamwise vortices, and the development of random noise. Linear disturbance simulations and stability diagnostics are also performed for a base flow consisting of the suction boundary layer and a streak. The scenarios are found to trigger transition by similar mechanisms as obtained for other flows. Transition at the lowest initial energy is provided by the oblique wave scenario for the considered Reynolds numbers 500, 800, and 1200. The Reynolds number dependence on the energy thresholds are determined for each scenario. The threshold scales like Re−2.6 for oblique transition and like Re−2.1 for transition initiated by streamwise vortices and random noise, indicating that oblique transition has the lowest energy threshold also for larger Reynolds numbers.

Large eddy simulations of spatially growing subsonic and supersonic turbulent round jets

This paper presents an application of large eddy simulations (LESs) using the filtered structure function model to spatially developing compressible round jets issuing from a perturbed upstream velocity profile close to a top hat. For centreline Mach number M = 0.9 and Reynolds number Re = 3600, the numerical solution compares satisfactorily against a forced-jet direct numerical simulation (DNS) and experimental data, both previously reported. High Reynolds number (Re = 36 000) ‘free’ jets at Mach 0.7 (case 1) and 1.4 (case 2) are studied. Here, an isotropic random white-noise perturbation is superposed on the upstream velocity. The Mach 0.7 jet has a convective Mach number of 0.35, and is weakly affected by compressibility. In this case, axisymmetric vortex rings are first shed from the nozzle and undergo alternate pairing further downstream. Then turbulence develops. The centreline velocity decay and some other statistical quantities are, in the self-similarity region, in very good agreement with previous incompressible experiments. At Mach 1.4, an impressive upstream reduction of the jet spreading rate is observed, due to an important delay of Kelvin–Helmholtz instability due to compressibility effects. Alternate pairing occurs immediately, and vortices are much more elongated in the flow direction. Further downstream, the jet becomes subsonic, develops into turbulence and spreads out again at a rate comparable with its subsonic counterpart. The potential-core length is increased by 27% from the subsonic to the supersonic case. This is in agreement with several laboratory experiments. Finally, the effects of Mach number increase upon various statistical quantities such as Reynolds stresses and radial lengthscale are studied. Results compare favourably against some experiments and temporal DNSs. From the point of view of Lumley's anisotropy invariant map evaluated on the whole physical domain, the Mach 0.7 jet is dominated by axisymmetric structures and the Mach 1.4 jet by streamwise perturbations.

Linear and Nonlinear Dynamics of Localized Disturbances in Three-dimensional Boundary Layers

Linear and nonlinear dynamics of localized disturbances in laminar incompressible two- and three-dimensional boundary layers was investigated using the temporal direct numerical simulation. Laminar base profiles were computed with a Runge-Kutta block-structured finite volume software package. The time evolution of localized disturbances have been computed using the periodical model in stream- and spanwise directions. Pseudospectral methods were applied for the space discretization of the incompressible Navier-Stokes equations. All stages of time-evolution of the disturbances from linear to nonlinear (saturated) stage were examined, turbulent velocity profiles and r.m.s. velocity fluctuations have been computed at saturated stages.

Direct simulation of turbulent supersonic boundary layers by an extended temporal approach

The present paper addresses the direct numerical simulation of turbulent zero-pressure-gradient boundary layers on a flat plate at Mach numbers 3, 4.5 and 6 with momentum-thickness Reynolds numbers of about 3000. Simulations are performed with an extended temporal direct numerical simulation (ETDNS) method. Assuming that the slow streamwise variation of the mean boundary layer is governed by parabolized Navier–Stokes equations, the equations solved locally in time with a temporal DNS are modified by a distributed forcing term so that the parabolized Navier–Stokes equations are recovered for the spatial average. The correct mean flow is obtained without a priori knowledge, the streamwise mean-flow evolution being approximated from its upstream history. ETDNS reduces the computational effort by up to two orders of magnitude compared to a fully spatial simulation.We present results for a constant wall temperature Tw chosen to be equal to its laminar adiabatic value, which is about 2.5 T∞, 4.4 T∞ and 7 T∞, respectively, where T∞ is the free-stream temperature for the three Mach numbers considered. The simulations are initialized with transition-simulation data or with re-scaled turbulent data at different parameters. We find that the ETDNS results closely match experimental mean-flow data. The van Driest transformed velocity profiles follow the incompressible law of the wall with small logarithmic regions.Of particular interest is the significance of compressibility effects in a Mach number range around the limit of M∞ ≃ 5, up to which Morkovin's hypothesis is believed to be valid. The results show that pressure dilatation and dilatational dissipation correlations are small throughout the considered Mach number range. On the other hand, correlations derived from Morkovin's hypothesis are not necessarily valid, as is shown for the strong Reynolds analogy.

Nonlinear equilibrium solutions in a three-dimensional boundary layer and their secondary instability

The observed nonlinear saturation of crossflow vortices in the DLR swept-plate transition experiment, followed by the onset of high-frequency signals, motivated us to compute nonlinear equilibrium solutions for this flow and investigate their instability to high-frequency disturbances. The equilibrium solutions are independent of receptivity, i.e. the way crossflow vortices are generated, and thus provide a unique characterization of the nonlinear flow prior to turbulence. Comparisons of these equilibrium solutions with experimental measurements exhibit strong similarities. Additional comparisons with results from the nonlinear parabolized stability equations (PSE) and spatial direct numerical simulations (DNS) reveal that the equilibrium solutions become unstable to steady, spatial oscillations with very long wavelengths following a bifurcation close to the leading edge. Such spatially oscillating solutions have been observed also in critical layer theory computations. The nature of the spatial behaviour is herein clarified and shown to be analogous to that encountered in temporal direct numerical simulations. We then employ Floquet theory to systematically study the dependence of the secondary, high-frequency instabilities on the saturation amplitude of the equilibrium solutions. With increasing amplitude, the most amplified instability mode can be clearly traced to spanwise inflectional shear layers that occur in the wake-like portions of the equilibrium solutions (Malik et al. 1994 call it ‘mode I’ instability). Both the frequency range and the eigenfunctions resemble recent experimental measurements of Kawakami et al. (1999). However, the lack of an explosive growth leads us to believe that additional self-sustaining processes are active at transition, including the possibility of an absolute instability of the high-frequency disturbances.

A nonlinear mechanism for receptivity of free-stream disturbances

Numerical experiments on the interaction of simple vortical free-stream disturbances with a laminar boundary layer are presented. Both spatial and temporal direct numerical simulations (DNS) have been performed for two types of free-stream disturbances. A linear and a nonlinear receptivity mechanism were identified. The nonlinear mechanism was found to force streaks inside the boundary layer similar to those found in experiments on free-stream turbulence and it performed equally well for disturbances elongated in the streamwise direction as for and oblique free-stream disturbances. The boundary layer response caused by the nonlinear mechanism was, depending on the initial disturbance energy, comparable to that of the linear mechanism, which was only efficient for free-stream streamwise vortices. A parameter study revealed that the wall normal velocity component of the free-stream disturbances is more important for the investigated receptivity mechanisms than the streamwise component. The identified boundary layer receptivity mechanism, in which three-dimensional disturbances in the free-stream continuously force streaks inside the boundary layer, may explain differences between experimental results and previously suggested theories for the origin of streaks in boundary layers subjected to free-stream turbulence.

Analysis of sound generated by free shear flows using direct numerical simulations

The sound produced by large-scale structures in free shear flows is investigated using direct numerical simulations (DNS). Both temporal and spatial DNS approaches are investigated, with special attention given to the former due to its lower computational cost. It is shown that the temporal approach can be used to investigate Mach-wave generation but not the general sound emitted by large-scale subsonic sources. Consequently, temporal DNS data of supersonic mixing layers undergoing transition to turbulence are analyzed. Two dominant Mach waves are found, each of which is identified with vortical features in the flow. The spatial DNS approach is then investigated to study the effect of the computational box size on the sound calculation. A large-scale model is used to show that the low-subsonic case is more restrictive in that sense than the supersonic one. The first case is further investigated by calculating the basic radiation emitted by axisymmetric jets, using the compact Lighthill and Mohring–Kambe formulations. Boundary corrections are developed and tested to reduce the box size effect and to improve the agreement between the two formulations. Finally, the effect of the retarded-time variation along the jet is discussed.

Streamwise vortices in Large-Eddy simulations of mixing layers

The formation of three-dimensional vortices in spatially-growing incompressible mixing layers is investigated in Large-Eddy Simulation at zero molecular viscosity, with the aid of the Filtered Structure Function subgrid-scale model proposed in Ducros et al. (1996, J. Fluid Mech., 326, pp. 1–36). Up to the second pairing at least, strong sensitivity to the nature of the random upstream perturbations and the spanwise size of the domain is observed. In addition to ‘unequal’ pairings between Kelvin-Helmholtz billows having undergone a different number of pairings and destructive ‘translative-pairing-and-tearing’ events, the two types of flow patterns obtained in the temporal Direct Numerical Simulations of Comte et al. (1992, Phys. Fluids A, 4, 2761–2778), are recovered, namely, highly three-dimensional vortex lattices undergoing local or helical pairings, as in the experiments of Chandrsuda et al. (1978, J. Fluid Mech., 85), or quasi-two-dimensional billows undergoing successive pairings while stretching streamwise vortices in between each other, as observed by Bernal and Roshko (1986, J. Fluid Mech., 170, 499–525). These streamwise vortices form in a succession of stages involving local roll-up and pairing, as conjectured by Lin and Corcos (1984, J. Fluid Mech., 141). Streamwise vortices are also found in a mixing layer formed within a solid-propellant rocket engine. At the wall of the outlet nozzle, they interact and possibly merge with Dean-Görtler-type vortices.

Modeling of nonparallel effects in temporal direct numerical simulations of compressible boundary-layer transition

One important alternative to spatial direct numerical simulation (SDNS) of a growing boundary-layer transition is a temporal direct numerical simulation (TDNS), where the flow is assumed to be locally parallel and the transition develops in time. To model nonparallel effects of a growing boundary layer, the TDNS allows the boundary layer to grow in time. This approach has been shown to be effective for an incompressible boundary layer. For a compressible boundary layer, however, a simple application of this approach has been found to be insufficient. To investigate this issue, we first split the variation of the flow field in the streamwise direction into a slowly evolving part and a fast and small-scale fluctuation part. By Taylor-expanding the slowly evolving large-scale part, this study shows that the Navier-Stokes operator can be reformulated as a power series of the perturbation parameter (x−x 0), yielding one set of equations for each power. Each set of these equations has a periodic solution in the streamwise direction, and therefore a modified TDNS method can be employed to solve these equations. Only the first set of the equations is considered in the applications presented. During the linear stage of transition, the results from this extended formulation show a significant improvement over those from the previous parallel flow formulation, especially for second modes which have short wavelengths. The results are well comparable with those from parabolized stability equations (PSE) and SDNS. A good agreement between this extended formulation and SDNS results is also demonstrated at the nonlinear stage.

Direct numerical simulation and data analysis of a Mach 4.5 transitional boundary-layer flow

This paper describes the creation by temporal direct numerical simulation and the analysis based on the Reynolds stress transport equations of a high quality data set that represents the laminar–turbulent transition of a high-speed boundary-layer flow. Following Pruett and Zang [Theoret. Comput. Fluid Dyn. 3, 345 (1992)], and with the help of algorithmic refinements, the evolution of an axial, Mach 4.5 boundary-layer flow along the exterior of a hollow cylinder is simulated numerically. From a perturbed laminar initial state, the well-resolved simulation proceeds through laminar breakdown to the beginning of a turbulent flow regime. Favre-averaged Reynolds stress transport equations are derived in generalized curvilinear coordinates and are then specialized to the cylindrical geometry at hand. Reynolds stresses and various turbulence quantities, such as turbulent kinetic energy and turbulent Mach number, are calculated from the numerical data at various stages of the transition process. The kinetic energy ‘‘budgets’’ are also constructed from the transport equations. Various contributing terms for the evolution of kinetic energy, like the rates of production, dissipation, transport, and diffusion, are presented. The compressible dissipation rate is small in comparison with the solenoidal dissipation rate for all times. The pressure–dilatation term is of the same order of magnitude as the compressible dissipation rate. The authors hope that both the data set and the analysis presented will benefit those who attempt to model high-speed transitional flow.

On the nonlinear stability of a high-speed, axisymmetric boundary layer

The stability of a high-speed, axisymmetric boundary-layer flow is investigated by means of secondary instability theory and direct numerical simulation. Parametric studies based on temporal secondary instability theory identify subharmonic secondary instability as a likely path to transition on a hollow cylinder at Mach 4.5. The theoretical predictions are validated by direct numerical solution of the compressible Navier–Stokes equations. Initial perturbations for the temporal direct numerical simulation consist of an axisymmetric ‘‘second-mode’’ primary disturbance and a subharmonic secondary disturbance comprised of four oblique wave components. At small initial amplitudes of the secondary disturbance, growth rates obtained from the spectrally accurate numerical simulation agree to several significant digits with linear growth rates predicted by secondary instability theory. Qualitative agreement persists to relatively large amplitudes of the secondary disturbance. Moderate transverse curvature is shown to significantly affect the growth rate of axisymmetric ‘‘second-mode’’ disturbances, the likely candidates of primary instability. The influence of curvature on secondary instability is largely indirect, but most probably significant, through modulation of the primary disturbance amplitude. Subharmonic secondary instability is shown to be predominantly inviscid in nature, and to account for peaks in the Reynolds stress components near the critical layer.