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The basic numerical techniques used for the direct simulation of transition to turbulence are described. Applications are presented for the computation of finite amplitude, two-dimensional steady states, the simulation of the secondary instability of channels and boundary-layers, and the simulation of the breakdown of detached shear layers.
Numerical simulations are used to investigate the relaxation of an unconfined, helically turbulent, fully three-dimensional magnetofluid, with conditions similar to those which are thought to result in the heating of the solar corona. In these simulations, the system evolves through a succession of force free states. After a relatively quiescent period of Ohmic decay, a phase of accelerated magnetic energy dissipation occurs. Some magnetic energy is transformed into kinetic energy, and the magnitude of entrophy created is a nontrival fraction of the mean square electric current. Concentrated vorticity structures are seen to play almost as important a role as electric current sheets in the heating process. Coincident with this accelerated dissipation process, a reorganization of the magnetic fields occurs, with transfer of magnetic energy to both shorter and longer wavelength modes than are initially present. The ratio of the magnetic field to the electric current density, alpha does not in general tend to assume a constant value in the force free regions during the evolution of the magnetofluid.
The restricted Euler equation captures many important features of the behavior of the velocity gradient tensor observed in direct numerical simulations (DNS) of isotropic turbulence. However, in slightly more complex flows the agreement is not good, especially in regions of low dissipation. In this paper, it is demonstrated that the Reynolds-averaged restricted Euler equation violates the balance of mean momentum for virtually all homogeneous turbulent flows with only two major exceptions: isotropic and homogeneously-sheared turbulence. A new model equation which overcomes this shortcoming and is more widely applicable is suggested. This modele is derived from the Navier-Stokes equation with a restricted Euler type approximation made on the fluctuating velocity gradient field. Analytical solutions of the proposed modified restricted Euler equation appear to be difficult to obtain. Hence, a strategy for numerically calculating the velocity gradient tensor is developed. Preliminary calculations tend to indicate that the modified restricted Euler equation captures many important aspects of the behavior of the fluctuating velocity gradients in anisotropic homogeneous turbulence.
To study the physics of small-scale properties of homogeneous isotropic turbulence at increasingly high Reynolds numbers, direct numerical simulation results have been obtained for forced isotropic turbulence at Taylor-scale Reynolds number R λ = 2500 on a 32 768 3 three-dimensional periodic domain using a GPU pseudo-spectral code on a 1.1 exaflop GPU supercomputer (Frontier). These simulations employ the multi-resolution independent simulation (MRIS) technique (Yeung & Ravikumar 2020, Phys. Rev. Fluids, vol. 5, 110517) where ensemble averaging is performed over multiple short segments initiated from velocity fields at modest resolution, and subsequently taken to higher resolution in both space and time. Reynolds numbers are increased by reducing the viscosity with the large-scale forcing parameters unchanged. Although MRIS segments at the highest resolution for each Reynolds number last for only a few Kolmogorov time scales, small-scale physics in the dissipation range is well captured – for instance, in the probability density functions and higher moments of the dissipation rate and enstrophy density, which appear to show monotonic trends persisting well beyond the Reynolds number range in prior works in the literature. Attainment of range of length and time scales consistent with classical scaling also reinforces the potential utility of the present high-resolution data for studies of short-time-scale turbulence physics at high Reynolds numbers where full-length simulations spanning many large-eddy time scales are still not accessible. A single snapshot of the 32 768 3 data is publicly available for further analyses via the Johns Hopkins Turbulence Database.
In this paper, we report a detailed experimental study of the role of ion-temperature-gradient driven turbulence in driving ion thermal transport in neutral beam injection (NBI)-heated L-mode plasmas in Experimental Advanced Superconducting Tokamak (EAST) (Wan et al 2000 Nucl. Fusion 40 1057) coupled with linear and nonlinear gyrokinetic simulations. Significant ion-scale turbulence k ⟂ < cm −1 (k ⟂ < 1.5, where k ⟂ where is the perpendicular wavenumber and ρ s is the ion gyroradius calculated using local electron temperature T e ), measured with a microwave reflectometer, is observed in the plasma core at for r/a ≈ 0.265 and 0.5 (where r is half the diameter of the closed flux surface at a given radial position, and a is half the diameter of the last closed flux surface). Local linear stability analysis with experimental equilibrium quantities at these two radial locations using the GS2 gyrokinetic code shows that the most unstable ion-scale micro-instability is the ion temperature gradient (ITG) mode. Since the computed maximum ion-scale linear growth rates are larger than the local Waltz-Miller E × B shearing rate (Waltz and Miller 1999 Phys. Plasmas 6 4265), the E × B shear is unable to suppress ion-scale turbulence, consistent with the experimental observation of ion-scale turbulence. Experimental ion and electron thermal transports, calculated with power balance analysis, are both anomalous with ion thermal transport being the dominant channel. Nonlinear gyrokinetic simulations using the electrostatic global particle-in-cell gyrokinetic tokamak simulation (GTS) code (Weixing Wang et al 2010 Phys. Plasmas 17 072511) show good quantitative agreement between predicted ion thermal transport and that from the power balance analysis. These results demonstrate for the first time the important role of ITG turbulence in driving thermal transport in NBI-heated L-mode plasmas in EAST, showing the effectiveness of GTS code in simulating electrostatic turbulence in EAST and the need of suppressing ITG turbulence to improve energy confinement in EAST.
The use of computational fluid dynamics in scramjet engine component development is widespread in the existing literature. Unfortunately, the quantification of model-form uncertainties is rarely addressed with anything other than sensitivity studies, requiring that the computational results be intimately tied to and calibrated against existing test data. This practice must be replaced with a formal uncertainty quantification process for computational fluid dynamics to play an expanded role in the system design, development, and flight certification process. Due to ground test facility limitations, this expanded role is believed to be a requirement by some in the test and evaluation community if scramjet engines are to be given serious consideration as a viable propulsion device. An effort has been initiated at the NASA Langley Research Center to validate several turbulence closure models used for Reynolds-averaged simulations of scramjet isolator flows. The turbulence models considered were the Menter BSL, Menter SST, Wilcox 1998, Wilcox 2006, and the Gatski-Speziale explicit algebraic Reynolds stress models. The simulations were carried out using the VULCAN computational fluid dynamics package developed at the NASA Langley Research Center. A procedure to quantify the numerical errors was developed to account for discretization errors in the validation process. This procedure utilized the grid convergence index defined by Roache as a bounding estimate for the numerical error. The validation data was collected from a mechanically back-pressured constant area (1 2 inch) isolator model with an isolator entrance Mach number of 2.5. As expected, the model-form uncertainty was substantial for the shock-dominated, massively separated flowfield within the isolator as evidenced by a 6 duct height variation in shock train length depending on the turbulence model employed. Generally speaking, the turbulence models that did not include an explicit stress limiter more closely matched the measured surface pressures. This observation is somewhat surprising, given that stress-limiting models have generally been developed to better predict shock-separated flows. All of the models considered also failed to properly predict the shape and extent of the separated flow region caused by the shock boundary layer interactions. However, the best performing models were able to predict the isolator shock train length (an important metric for isolator operability margin) to within 1 isolator duct height.
Simulation experiments reveal key processes that organize a hydrostatic environment conducive to severe turbulence. The paradigm requires juxtaposition of the entrance region of a curved jet stream, which is highly subgeostrophic, with the entrance region of a straight jet stream, which is highly supergeostrophic. The wind and mass fields become misphased as the entrance regions converge resulting in the significant spatial variation of inertial forcing, centripetal forcing, and along- and cross-stream pressure gradient forcing over a mesobeta scale region. This results in frontogenesis and the along-stream divergence of cyclonic and convergence of cyclonic ageostrophic vertical vorticity. The centripetally forced mesoscale front becomes the locus of large gradients of ageostrophic vertical vorticity along an overturning isentrope. This region becomes favorable for streamwise vorticity gradient formation enhancing the environment for organization of horizontal vortex tubes in the presence of buoyant forcing.
Effects on shear layers in simulated confined and unconfined flows studied. Report presents comparative study of some terms that correct for effects of compressibility in standard k-epsilon mathematical model of turbulence where k denotes turbulence kinetic energy and epsilon denotes rate of dissipation of turbulence kenetic energy. Involved simulation of flows by numerical solution of Reynolds-averaged Navier-Stokes equations.
The turbulent kinetic energy transport equation, which is the basis of turbulence models, is investigated for homogeneous, compressible turbulence using direct numerical simulations performed at CTR. It is shown that the partition between dilatational and solenoidal modes is very sensitive to initial conditions for isotropic decaying turbulence but not for sheared flows. The importance of the dilatational dissipation and of the pressure-dilatation term is evidenced from simulations and a transport equation is proposed to evaluate the pressure-dilatation term evolution. This transport equation seems to work well for sheared flows but does not account for initial condition sensitivity in isotropic decay. An improved model is proposed.
Understanding the physics of low-confinement (L-), improved-confinement (I-), and high-confinement (H-) modes is critical for fusion reactors. The finding herein reports observations of two types of turbulence coexisting near the L-mode edge, one magnetohydrodynamic (MHD)-like and another micro-tearing mode (MTM)-like, linked to the H-mode and I-mode confinement in the DIII-D tokamak. Ion-scale magnetic and density turbulence is measured using a Faraday-effect radial-interferometer-polarimeter and beam-emission-spectroscopy (BES). Broadband turbulence spectra of up to ~600 kHz are observed in two discharges where transitions between L-mode, I-mode, and H-mode occurs. Turbulence is found to be inversely correlated with confinement, meaning lower turbulence power at higher confinement. Distinctively, the high-frequency (HF, >~100 kHz) magnetic turbulence power changes by the most (55%) during transitions primarily involving energy confinement change, whereas the low-frequency (LF, <~100 kHz) magnetic and density turbulence power changes by the most (80%) during transitions primarily involving particle confinement change. The LF turbulence amplitude oscillates with and leads to deuterium-alpha emission oscillations before an H-mode. These results imply that HF turbulence mainly affects energy confinement whereas LF turbulence can affect particle confinement. The magnetic and density turbulence exhibits coherence up to 0.6 and cross-phase magnitude close to π/2 in most cases, suggesting they have a common origin in both the LF and HF ranges. BES suggests that LF turbulence resides at the edge (ρ = 0.95) and HF turbulence can be at the outer core (ρ = 0.8) or edge (ρ = 0.95). Comparisons of measurements, theory, and gyrokinetic simulations suggest that HF turbulence is MTM-like in all cases, whereas LF turbulence is more consistent with MHD-like modes and the exact instability might change during transitions—except that a drift-wave origin is possible in a low collisionality H-mode. These results suggest that the H-mode involves suppressed MHD-like turbulence, whereas the I-mode mitigates MTM-like turbulence along with largely unchanged MHD-like turbulence.
Renewed interest in hypersonic propulsion systems has led to research programs investigating combined cycle engines that are designed to operate efficiently across the flight regime. The Rocket-Based Combined Cycle Engine is a propulsion system under development at the NASA Lewis Research Center. This engine integrates a high specific impulse, low thrust-to-weight, airbreathing engine with a low-impulse, high thrust-to-weight rocket. From takeoff to Mach 2.5, the engine operates as an air-augmented rocket. At Mach 2.5, the engine becomes a dual-mode ramjet; and beyond Mach 8, the rocket is turned back on. One Rocket-Based Combined Cycle Engine variation known as the "Strut-Jet" concept is being investigated jointly by NASA Lewis, the U.S. Air Force, Gencorp Aerojet, General Applied Science Labs (GASL), and Lockheed Martin Corporation. Work thus far has included wind tunnel experiments and computational fluid dynamics (CFD) investigations with the NPARC code. The CFD method was initiated by modeling the geometry of the Strut-Jet with the GRIDGEN structured grid generator. Grids representing a subscale inlet model and the full-scale demonstrator geometry were constructed. These grids modeled one-half of the symmetric inlet flow path, including the precompression plate, diverter, center duct, side duct, and combustor. After the grid generation, full Navier-Stokes flow simulations were conducted with the NPARC Navier-Stokes code. The Chien low-Reynolds-number k-e turbulence model was employed to simulate the high-speed turbulent flow. Finally, the CFD solutions were postprocessed with a Fortran code. This code provided wall static pressure distributions, pitot pressure distributions, mass flow rates, and internal drag. These results were compared with experimental data from a subscale inlet test for code validation; then they were used to help evaluate the demonstrator engine net thrust.
In response to the global SARS-CoV-2 transmission pandemic, Sandia National Laboratories Rapid Lab-Directed Research and Development (LDRD) COVID-19 initiative has deployed a multi-physics, droplet-laden, turbulent low-Mach simulation tool to model pathogen-containing water droplets that emanate from synthetic human coughing and breathing. The low-Mach turbulent Eulerian/point-particle Lagrangian methodology directly couples mass, momentum, energy, and species to capture droplet evaporation physics that supports the ability to distinguish between droplets that deposit and those that persist in the environment. Additionally, the cough mechanism is modeled as a pulsed-spray with a prescribed log-normal droplet size distribution. Simulations demonstrate direct droplet deposition lengths in excess of three meters while the persistence of a droplet nuclei entrained within a buoyant plume is noted. Including the effect of protective barriers demonstrates effective mitigation of large-droplet transport. For coughs into a protective barrier, jet impingement and large-scale re-circulation can drive droplets vertically and back towards the subject while supporting persistence of droplet nuclei. Simulations in quiescent conditions demonstrate droplet preferential concentrations due to the coupling between vortex ring shedding and the subsequent advection of a series of three-dimensional rings that tilt and rise vertically due to a misalignment between the initial principle vortex trajectory and gravity. These resolved coughing simulations note vortex ring formation, roll-up and breakdown, while entraining droplet nuclei for large distances and time scales.
Developed a new algorithm for the simulation of three dimensional homogeneous turbulent velocity fields. For typical atmospheric conditions it is impossible to produce a simulated velocity field that simultaneously satisfy a given spatial correlation and the corresponding spatial spectrum because of spectral aliasing. The new algorithms produce a turbulent velocity field which has accurate spatial correlations which is required for performance predictions from space-based systems. Developed a new algorithm for extracting the spatial statistics of the atmospheric velocity field using coherent Doppler lidar. The performance of the algorithm was compared with past methods and the new algorithm produces useful results for space-based data, which was not possible before. Developed new methods for verification of the errors in ground-based and space-based Doppler lidar wind measurements. These new methods do not require independent in situ data. This is an important issue for the verification of space-based Doppler lidar measurements of the global wind field. The performance of the new algorithm was compared with past results for both space-based and ground-based operation. The new algorithm has the best performance and is the only algorithm that performed satisfactory for spacebased operation. The performance of coherent Doppler lidar for a space missions with various scanning geometries was determined using computer simulation which contained the effects of random instrumental velocity errors, wind shear, wind variability along the range-gate and from shot-to-shot, and random variations in atmospheric aerosol backscatter over the measurement volume. The bias in the velocity estimates was small and the accuracy in the is typically less than 0.5 m/s for high signal conditions. For a large number of shot per velocity estimate, the threshold signal level for acceptable estimates is proportional to the number of shots to the minus one half power. This agrees with previous results determined for ground-based measurements. The use of multi-element optical detectors for autonomous operation of coherent Doppler lidar was shown to be a very promising technique. Optimal detector geometries were determined by computer simulation of performance: for ground-based testing with a fixed calibration target and for space-based operation using the random surface returns. The effects of refractive turbulence on ground-based calibration of coherent Doppler lidar was determined by computer simulations and compared with theoretical predictions. New techniques were required to correctly predict performance for the focused beam geometry commonly used for verification of space-based operation. An improved velocity estimator was evaluated for space-based applications were signal shot measurements are used to produce vector wind measurements. This permits more accurate measurements when the signal level is not known a priori or not available from multiple shot measurements. The average Doppler lidar signal spectrum including the effects of velocity turbulence was derived and calculated. This permits new estimation algorithms for turbulence based on spectral estimates. In situ atmospheric measurements were conducted and analyzed using an instrumented kite-platform. This work helps provide the required in situ data for verification of Doppler lidar velocity statistics.
This second part of our research explores the kinematic aspect of fine-scale bubble-induced turbulence (BIT) to (i) present the effect of bubble breakup and coalescence and (ii) compare it against the universal kinematic fine-scale turbulence characteristics reported in the literature. To this end, we simulate a dilute bubbly system of 0.5% void fraction using two distinct numerical simulations. In the volume-of-fluid (VoF) simulation, bubbles undergo breakup and coalescence. In the immersed boundary method (IBM) simulation, however, they act as rigid spheres. We also perform a simulation of classical homogeneous isotropic turbulence (HIT). The first important outcome of this study is that BIT is radically different from HIT in terms of its kinematic fine-scale characteristics. In the vorticity-dominating regions, BIT exhibits a weak vortex stretching. This weak vortex stretching is due to (a) the intermediate strain-rate eigenvalues skewed weakly to positive and (b) the extensive strain-rate eigenvector aligning perpendicular to the vorticity vector. The BIT has, on average, not only a weak enstrophy production but also a weak strain production in strain-dominating regions. The weak strain production is due to (a) the presence of vortex stretching in highly strained fluid elements and (b) the absolute magnitude of compressive strain-rate eigenvalue being as close to the extensive strain-rate eigenvalue. Thus, none of the kinematic fine-scale HIT characteristics is noted for BIT. The second important conclusion is that bubble breakup and coalescence play little to no influence on the kinematics of fine-scale BIT as VoF and IBM simulations produce similar results.
The two-dimensional Reynolds-averaged compressible Navier-Stokes equations are solved using MacCormack's second-order-accurate explicit finite difference method to simulate the separated transonic turbulent flow field over an airfoil. Four different algebraic eddy viscosity models are tested for viability to achieve turbulence closure for the class of flows considered. These models range from an unmodified boundary-layer mixing-length model to a relaxation model incorporating special considerations for the separation bubble region. Results of this study indicate the necessity for special attention to the separated flow region and suggest limits of applicability of algebraic turbulence models to these separated flow fields.
A theoretical interpretation of sounding rocket measurements of intense Langmuir wave fields (less than or equal to 500 mV/m) driven by a stream of 20 eV to 4 keV electrons in the lower auroral zone is developed. This interpretation is based on the ability of the 10 microseconds sampling rate of the wave detector to temporally resolve the structure of the Langmuir wave field envelope. A modified form of the Zakharov equations is used to numerically study beam-driven Langmuir turbulence in the presence of a moderate magnetic field (OMEGA (sub e) approximately equals Omega (sub pe). Strong Landau damping on observed nonthermal scattered electrons, which is treated in a companion paper (Newman et al., this issue), plays an important role by inhibiting backscatter cascade and the development of strong turbulence. A parameterized model of the linear electron stream-driven wave instability is introduced, which incorporates limited quasilinear plateau formation. A reasonable set of parameters is found that yields semiquantitative agreement between observed properties of the Langmuir fields and the results of Zakharov equation simulations, including the amplitude and characteristic frequency of the electric field envelope modulations.
Direct numerical simulation at 256(exp 3) resolution have been carried out to study the response of isotropic turbulence to the concurrent effects of solid-body rotation and numerical forcing at the large scales. Because energy transfer to the smaller scales is weakened by rotation, energy input from forcing gradually builds up at the large scales, causing the overall kinetic energy to increase. At intermediate wavenumbers the energy spectrum undergoes a transition from a limited k(exp -5/3) inertial range to k(exp -2) scaling recently predicted in the literature. Although the Reynolds stress tensor remains approximately isotropic and three-components, evidence for anisotropy and quasi- two-dimensionality in length scales and spectra in different velocity components and directions is strong. The small scales are found to deviate from local isotropy, primarily as a result of anisotropic transfer to the high wavenumbers. To understand the spectral dynamics of this flow we study the detailed behavior of nonlinear triadic interactions in wavenumber space. Spectral transfer in the velocity component parallel to the axis of rotation is qualitatively similar to that in non-rotating turbulence; however the perpendicular component is characterized by a greatly suppressed energy cascade at high wavenumber and a local reverse transfer at the largest scales. The broader implications of this work are briefly addressed.
A filtered density function (FDF) method suitable for chemically reactive flows is developed in the context of large eddy simulation. The advantage of the FDF methodology is its inherent ability to resolve subgrid scales (SGS) scalar correlations that otherwise have to be modeled. Because of the lack of robust models to accurately predict these correlations in turbulent reactive flows, simulations involving turbulent combustion are often met with a degree of skepticism. The FDF methodology avoids the closure problem associated with these terms and treats the reaction in an exact manner. The scalar FDF approach is particularly attractive since it can be coupled with existing hydrodynamic computational fluid dynamics (CFD) codes.