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NASA NTRS1991

The use of direct numerical simulation data in turbulence modeling

Direct numerical simulations (DNS) of turbulent flows provide a complete data base to develop and to test turbulence models. In this article, the progress made in developing models for the dissipation rate equation is reviewed. New scaling arguments for the various terms in the dissipation rate equation were tested using data from DNS of homogeneous shear flows. Modifications to the epsilon-equation model that take into account near-wall effects were developed using DNS of turbulent channel flows. Testing of new models for flows under mean compression was carried out using data from DNS of isotropically compressed turbulence. In all of these studies the data from the simulations was essential in guiding the model development. The next generation of DNS will be at higher Reynolds numbers, and will undoubtedly lead to improved models for computations of flows of practical interest.

Mansour, N. N.↗

NASA NTRS1993

The combustion program at CTR

Understanding and modeling of turbulent combustion are key problems in the computation of numerous practical systems. Because of the lack of analytical theories in this field and of the difficulty of performing precise experiments, direct numerical simulation (DNS) appears to be one of the most attractive tools to use in addressing this problem. The general objective of DNS of reacting flows is to improve our knowledge of turbulent combustion but also to use this information for turbulent combustion models. For the foreseeable future, numerical simulation of the full three-dimensional governing partial differential equations with variable density and transport properties as well as complex chemistry will remain intractable; thus, various levels of simplification will remain necessary. On one hand, the requirement to simplify is not necessarily a handicap: numerical simulations allow the researcher a degree of control in isolating specific physical phenomena that is inaccessible in experiments. CTR has pursued an intensive research program in the field of DNS for turbulent reacting flows since 1987. DNS of reacting flows is quite different from DNS of non-reacting flows: without reaction, the equations to solve are clearly the five conservation equations of the Navier Stokes system for compressible situations (four for incompressible cases), and the limitation of the approach is the Reynolds number (or in other words the number of points in the computation). For reacting flows, the choice of the equations, the species (each species will require one additional conservation equation), the chemical scheme, and the configuration itself is more complex.

Poinsot, Thierry J.↗

NASA NTRS1994

Direct numerical simulations and modeling of a spatially-evolving turbulent wake

Understanding of turbulent free shear flows (wakes, jets, and mixing layers) is important, not only for scientific interest, but also because of their appearance in numerous practical applications. Turbulent wakes, in particular, have recently received increased attention by researchers at NASA Langley. The turbulent wake generated by a two-dimensional airfoil has been selected as the test-case for detailed high-resolution particle image velocimetry (PIV) experiments. This same wake has also been chosen to enhance NASA's turbulence modeling efforts. Over the past year, the author has completed several wake computations, while visiting NASA through the 1993 and 1994 ASEE summer programs, and also while on sabbatical leave during the 1993-94 academic year. These calculations have included two-equation (K-omega and K-epsilon) models, algebraic stress models (ASM), full Reynolds stress closure models, and direct numerical simulations (DNS). Recently, there has been mutually beneficial collaboration of the experimental and computational efforts. In fact, these projects have been chosen for joint presentation at the NASA Turbulence Peer Review, scheduled for September 1994. DNS calculations are presently underway for a turbulent wake at Re(sub theta) = 1000 and at a Mach number of 0.20. (Theta is the momentum thickness, which remains constant in the wake of a two dimensional body.) These calculations utilize a compressible DNS code written by M. M. Rai of NASA Ames, and modified for the wake by J. Cimbala. The code employs fifth-order accurate upwind-biased finite differencing for the convective terms, fourth-order accurate central differencing for the viscous terms, and an iterative-implicit time-integration scheme. The computational domain for these calculations starts at x/theta = 10, and extends to x/theta = 610. Fully developed turbulent wake profiles, obtained from experimental data from several wake generators, are supplied at the computational inlet, along with appropriate noise. After some adjustment period, the flow downstream of the inlet develops into a fully three-dimensional turbulent wake. Of particular interest in the present study is the far wake spreading rate and the self-similar mean and turbulence profiles. At the time of this writing, grid resolution studies are underway, and a code is being written to calculate turbulence statistics from these wake calculations; the statistics will be compared to those from the ongoing PIV wake measurements, those of previous experiments, and those predicted by the various turbulence models. These calculations will lead to significant long-term benefits for the turbulence modeling effort. In particular, quantities such as the pressure-strain correlation and the dissipation rate tensor can be easily calculated from the DNS results, whereas these quantities are nearly impossible to measure experimentally. Improvements to existing turbulence models (and development of new models) require knowledge about flow quantities such as these. Present turbulence models do a very good job at prediction of the shape of the mean velocity and Reynolds stress profiles in a turbulent wake, but significantly underpredict the magnitude of the stresses and the spreading rate of the wake. Thus, the turbulent wake is an ideal flow for turbulence modeling research. By careful comparison and analysis of each term in the modeled Reynolds stress equations, the DNS data can show where deficiencies in the models exist; improvements to the models can then be attempted.

Cimbala, John M.↗

NASA NTRS1999

A Priori Subgrid Analysis of Temporal Mixing Layers with Evaporating Droplets

Subgrid analysis of a transitional temporal mixing layer with evaporating droplets has been performed using three sets of results from a Direct Numerical Simulation (DNS) database, with Reynolds numbers (based on initial vorticity thickness) as large as 600 and with droplet mass loadings as large as 0.5. In the DNS, the gas phase is computed using a Eulerian formulation, with Lagrangian droplet tracking. The Large Eddy Simulation (LES) equations corresponding to the DNS are first derived, and key assumptions in deriving them are first confirmed by computing the terms using the DNS database. Since LES of this flow requires the computation of unfiltered gas-phase variables at droplet locations from filtered gas-phase variables at the grid points, it is proposed to model these by assuming the gas-phase variables to be the sum of the filtered variables and a correction based on the filtered standard deviation; this correction is then computed from the Subgrid Scale (SGS) standard deviation. This model predicts the unfiltered variables at droplet locations considerably better than simply interpolating the filtered variables. Three methods are investigated for modeling the SGS standard deviation: the Smagorinsky approach, the Gradient model and the Scale-Similarity formulation. When the proportionality constant inherent in the SGS models is properly calculated, the Gradient and Scale-Similarity methods give results in excellent agreement with the DNS.

Okongo, Nora↗

NASA NTRS2000

Towards LES Models of Jets and Plumes

As pointed out by Rodi standard integral solutions for jets and plumes developed for discharge into infinite, quiescent ambient are difficult to extend to complex situations, particularly in the presence of boundaries such as the sea floor or ocean surface. In such cases the assumption of similarity breaks down and it is impossible to find a suitable entrainment coefficient. The models are also incapable of describing any but the most slowly varying unsteady motions. There is therefore a need for full time-dependent modeling of the flow field for which there are three main approaches: (1) Reynolds averaged numerical simulation (RANS), (2) large eddy simulation (LES), and (3) direct numerical simulation (DNS). Rodi applied RANS modeling to both jets and plumes with considerable success, the test being a match with experimental data for time-averaged velocity and temperature profiles as well as turbulent kinetic energy and rms axial turbulent velocity fluctuations. This model still relies on empirical constants, some eleven in the case of the buoyant jet, and so would not be applicable to a partly laminar plume, may have limited use in the presence of boundaries, and would also be unsuitable if one is after details of the unsteady component of the flow (the turbulent eddies). At the other end of the scale DNS modeling includes all motions down to the viscous scales. Boersma et al. have built such a model for the non-buoyant case which also compares well with measured data for mean and turbulent velocity components. The model demonstrates its versatility by application to a laminar flow case. As its name implies, DNS directly models the Navier-Stokes equations without recourse to subgrid modeling so for flows with a broad spectrum of motions (high Re) the cost can be prohibitive - the number of required grid points scaling with Re(exp 9/4) and the number of time steps with Re(exp 3/4). The middle road is provided by LES whereby the Navier-Stokes equations are formally filtered with the filter chosen to only exclude the smallest turbulent motions. If successful, LES should provide much of the detail available to DNS but at more bearable cost. Fatica et al. in comparing LES with DNS for a low Reynolds number jet showed that the LES could simulate the temporally evolving behavior including growth of the jet thickness. It is the intention of this report to explore the application of an LES model to jets and plumes. As always, before tackling complex situations, the model must be tested for the simplest of cases and so we address only two, a non-buoyant axisymmetric jet issuing steadily from an orifice into a semi-infinite stationary environment and a buoyant jet in the same environment. The work is a continuation of Basu and Mansour.

Webb, A. T.↗

NASA NTRS2002

An Evaluation of Linear Instability Waves as Sources of Sound in a Supersonic Turbulent Jet

Mach wave radiation from supersonic jets is revisited to better quantify the extent to which linearized equations represent the details of the actual mechanism. To this end, we solve the linearized Navier-Stokes equations (LNS) with precisely the same mean flow and inflow disturbances as a previous direct numerical simulation (DNS) of a perfectly expanded turbulent M = 1.92 jet. We restrict our attention to the first two azimuthal modes, n = 0 and n = 1, which constitute most of the acoustic field. The direction of peak radiation and the peak Strouhal number matches the DNS reasonably well, which is in accord with previous experimental justification of the linear theory. However, it is found that the sound pressure level predicted by LNS is significantly lower than that from DNS. In order to investigate the discrepancy, individual frequency components of the solution are examined. These confirm that near the peak Strouhal number, particularly for the first helical mode n = 1, the amplification of disturbances in the LNS closely matches the DNS. However, away from the peak frequency (and generally for the azimuthal mode n = 0), modes in the LNS are damped while those in the DNS grow at rates comparable to those at the peak Strouhal number.

Mohseni, Kamran↗

NASA NTRS2005

Consistent Large-Eddy Simulation of a Temporal Mixing Layer Laden with Evaporating Drops: A Posteriori Modelling - Part 2

Large-eddy simulation (LES) is conducted of a three-dimensional temporal mixing layer whose lower stream is initially laden with liquid drops which may evaporate during the simulation. The gas-phase equations are written in an Eulerian frame for two perfect gas species (carrier gas and vapour emanating from the drops), while the liquid-phase equations are written in a Lagrangian frame. The effect of drop evaporation on the gas phase is considered through mass, species, momentum and energy source terms. The drop evolution is modelled using physical drops, or using computational drops to represent the physical drops. Simulations are performed using various LES models previously assessed on a database obtained from direct numerical simulations (DNS). These LES models are for: (i) the subgrid-scale (SGS) fluxes and (ii) the filtered source terms (FSTs) based on computational drops. The LES, which are compared to filtered-and-coarsened (FC) DNS results at the coarser LES grid, are conducted with 64 times fewer grid points than the DNS, and up to 64 times fewer computational than physical drops. It is found that both constant-coefficient and dynamic Smagorinsky SGS-flux models, though numerically stable, are overly dissipative and damp generated small-resolved-scale (SRS) turbulent structures. Although the global growth and mixing predictions of LES using Smagorinsky models are in good agreement with the FC-DNS, the spatial distributions of the drops differ significantly. In contrast, the constant-coefficient scale-similarity model and the dynamic gradient model perform well in predicting most flow features, with the latter model having the advantage of not requiring a priori calibration of the model coefficient. The ability of the dynamic models to determine the model coefficient during LES is found to be essential since the constant-coefficient gradient model, although more accurate than the Smagorinsky model, is not consistently numerically stable despite using DNS-calibrated coefficients. With accurate SGS-flux models, namely scale-similarity and dynamic gradient, the FST model allows up to a 32-fold reduction in computational drops compared to the number of physical drops, without degradation of accuracy; a 64-fold reduction leads to a slight decrease in accuracy.

two-phase flow↗

NASA NTRS2004

Part 1 of a Computational Study of a Drop-Laden Mixing Layer

This first of three reports on a computational study of a drop-laden temporal mixing layer presents the results of direct numerical simulations (DNS) of well-resolved flow fields and the derivation of the large-eddy simulation (LES) equations that would govern the larger scales of a turbulent flow field. The mixing layer consisted of two counterflowing gas streams, one of which was initially laden with evaporating liquid drops. The gas phase was composed of two perfect gas species, the carrier gas and the vapor emanating from the drops, and was computed in an Eulerian reference frame, whereas each drop was tracked individually in a Lagrangian manner. The flow perturbations that were initially imposed on the layer caused mixing and eventual transition to turbulence. The DNS database obtained included transitional states for layers with various liquid mass loadings. For the DNS, the gas-phase equations were the compressible Navier-Stokes equations for conservation of momentum and additional conservation equations for total energy and species mass. These equations included source terms representing the effect of the drops on the mass, momentum, and energy of the gas phase. From the DNS equations, the expression for the irreversible entropy production (dissipation) was derived and used to determine the dissipation due to the source terms. The LES equations were derived by spatially filtering the DNS set and the magnitudes of the terms were computed at transitional states, leading to a hierarchy of terms to guide simplification of the LES equations. It was concluded that effort should be devoted to the accurate modeling of both the subgridscale fluxes and the filtered source terms, which were the dominant unclosed terms appearing in the LES equations.

Okong'o, Nora A.↗

NASA NTRS2013

Studies of Inviscid Flux Schemes for Acoustics and Turbulence Problems

Five different central difference schemes, based on a conservative differencing form of the Kennedy and Gruber skew-symmetric scheme, were compared with six different upwind schemes based on primitive variable reconstruction and the Roe flux. These eleven schemes were tested on a one-dimensional acoustic standing wave problem, the Taylor-Green vortex problem and a turbulent channel flow problem. The central schemes were generally very accurate and stable, provided the grid stretching rate was kept below 10%. As near-DNS grid resolutions, the results were comparable to reference DNS calculations. At coarser grid resolutions, the need for an LES SGS model became apparent. There was a noticeable improvement moving from CD-2 to CD-4, and higher-order schemes appear to yield clear benefits on coarser grids. The UB-7 and CU-5 upwind schemes also performed very well at near-DNS grid resolutions. The UB-5 upwind scheme does not do as well, but does appear to be suitable for well-resolved DNS. The UF-2 and UB-3 upwind schemes, which have significant dissipation over a wide spectral range, appear to be poorly suited for DNS or LES.

Morris, Chris↗

NASA NTRS2018

Numerical Study of Turbulent Separation Bubbles with Varying Pressure Gradient and Reynolds Number

A family of cases each containing a small separation bubble is treated by direct numerical simulation (DNS), varying two parameters: the severity of the pressure gradients, generated by suction and blowing across the opposite boundary, and the Reynolds number. Each flow contains a well-developed entry region with essentially zero pressure gradient, and all are adjusted to have the same value for the momentum thickness, extrapolated from the entry region to the centre of the separation bubble. Combined with fully defined boundary conditions this will make comparisons with other simulations and turbulence models rigorous; we present results for a set of eight Reynolds-averaged Navier–Stokes turbulence models. Even though the largest Reynolds number is approximately 5.5 times higher than in a similar DNS study we presented in 1997, the models have difficulties matching the DNS skin friction very closely even in the zero pressure gradient, which complicates their assessment. In the rest of the domain, the separation location per se is not particularly difficult to predict, and the most definite disagreement between DNS and models is near reattachment. Curiously, the better models tend to cluster together in their predictions of pressure and skin friction even when they deviate from the DNS, although their eddy-viscosity levels are widely different in the outer region near the bubble (or they do not rely on an eddy viscosity). Stratford’s square-root law is satisfied by the velocity profiles, both at separation and reattachment. The Reynolds-number range covers a factor of two, with the Reynolds number based on the extrapolated momentum thickness equal to approximately 1500 and 3000. This allows tentative estimates of the improvements that even higher values will bring to the model comparisons. The solutions are used to assess models through pressure, skin friction and other measures; the flow fields are also used to produce effective eddy-viscosity targets for the models, thus guiding turbulence-modelling work in each region of the flow.

Coleman, G. N.↗

DOE OSTI2020

A priori analysis of a power-law sub-filter scale mixing model for transported PDF modeling of high Karlovitz turbulent premixed flames

Accurate modeling of mixing in large-eddy simulation (LES) / transported probability density function (PDF) modeling of turbulent combustion remains an outstanding issue. The issue is particularly salient in turbulent premixed combustion under extreme conditions such as high-Karlovitz number Ka. Here, the present study addresses this issue by conducting an a priori analysis of a power-law scaling based mixing timescale model for the transported PDF model. A recently produced DNS dataset of a high-Ka turbulent jet flame is used for the analysis. A power-law scaling is observed for a scaling factor used to model the sub-filter scale mixing timescale in this high-Ka turbulent premixed DNS flame when the LES filter size is much greater than the characteristic thermal thickness of a laminar premixed flame. The sensitivity of the observed power-law scaling to the different viewpoints (local or global) and to the different scalars for the data analysis is examined and the dependence of the model parameters on the dimensionless numbers Ka and Re (the Reynolds number) is investigated. Different model formulations for the mixing timescale are then constructed and assessed in the DNS flame. The proposed model is found to be able to reproduce the mixing timescale informed by the high-Ka DNS flame significantly better than a previous model.

42 ENGINEERING↗

NASA NTRSDate not supplied

DeepONet-Assisted Optimization of Surface Topography for Transition Delay in a Mach 4.5 Boundary Layer

We use deep learning, an ensemble variational technique (EnVar), and direct numerical simulations(DNS) to design an optimal topography for a two-dimensional roughness element that delays the on-set of laminar-turbulent transition in a Mach 4.5 flat-plate boundary layer. Deep operator networks (DeepONets), which have the known ability to learn complex nonlinear operators within dynamical systems, are used for machine learning. For the baseline configuration of a smooth flat plate, the second-mode waves at the DNS inflow cause a quick nonlinear breakdown of the high-speed boundary layer within the computational domain. Results reported in the present study validate the ability of DeepONets to model the transition delay via a given topography of the roughness element. The computing cost to optimize the rough-ness element for minimal skin-friction drag is substantially lowered by the DeepONets-based reduced-order model. In comparison to the baseline method of EnVar optimization based on DNS alone, the DeepONets-based EnVar optimizer is able to delay transition past the outflow boundary of the computational domain while utilizing almost 5–6 times fewer DNS.

Machine Learning↗

DOE OSTI2023

New scaling law for turbulent boundary layers with high surface mass transfer

A fundamental understanding of the phenomena occurring in the turbulent boundary layer in the presence of surface blowing is limited, and considerable disagreements persist even in describing primary quantities, such as the boundary layer profile. The theories based on the linear boundary layer equations show that the thickness of the sublayer increases in the presence of surface blowing; therefore, the viscous sublayer and law of the wall modify. In this study, direct numerical simulations (DNS) of turbulent boundary layers with uniform surface mass transfers are carried out in order to scale the velocity profile. Emphasis is placed on moderate to high mass transfer rates, which are relevant to the most common hybrid rockets configuration. DNS data are used to establish a functional law of the wall and a law of wake by means of the relation between the wall shear stress and surface mass transfer. Analysis of the mean kinetic energy budget shows that the magnitude of turbulent kinetic energy increases by surface mass transfer, and the production rate extends significantly in the inner layer as the injection rate increases. DNS data of various surface blowing are used to complete the closure of turbulence kinetic energy equation and develop an eddy viscosity model. The predicted turbulent kinetic energy and eddy viscosity agree with DNS data for moderate to high blowing rates.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1990

Large-eddy simulation of transitional channel flow

A large-eddy simulation (LES) of transition in plane channel flow was carried out. The LES results were compared with those of a fine direct numerical simulation (DNS), and with those of a coarse DNS that uses the same mesh as the LES, but does not use a residual stress model. While at the early stages of transition, LES and coarse DNS give the same results: the presence of the residual stress model was found to be necessary to predict accurately mean velocity and Reynolds stress profiles during the late stages of transition (after the second spike stage). The evolution of single Fourier modes is also predicted more accurately by the LES than by the DNS. As small scales are generated, the dissipative character of the residual stress starts to reproduce correctly the energy cascade. As transition progresses, the flow approaches its fully developed turbulent state, the subgrid scales tend towards equilibrium, and the model becomes more accurate.

Piomelli, Ugo↗

NASA NTRS1992

Large eddy simulations and direct numerical simulations of high speed turbulent reacting flows

The basic objective of this research is to extend the capabilities of Large Eddy Simulations (LES) and Direct Numerical Simulations (DNS) for the computational analyses of high speed reacting flows. In the efforts related to LES, we were primarily involved with assessing the performance of the various modern methods based on the Probability Density Function (PDF) methods for providing closures for treating the subgrid fluctuation correlations of scalar quantities in reacting turbulent flows. In the work on DNS, we concentrated on understanding some of the relevant physics of compressible reacting flows by means of statistical analysis of the data generated by DNS of such flows. In the research conducted in the second year of this program, our efforts focused on the modeling of homogeneous compressible turbulent flows by PDF methods, and on DNS of non-equilibrium reacting high speed mixing layers. Some preliminary work is also in progress on PDF modeling of shear flows, and also on LES of such flows.

Givi, P.↗

NASA NTRS1992

Validation of three-dimensional incompressible spatial direct numerical simulation code: A comparison with linear stability and parabolic stability equation theories for boundary-layer transition on a flat plate

Spatially evolving instabilities in a boundary layer on a flat plate are computed by direct numerical simulation (DNS) of the incompressible Navier-Stokes equations. In a truncated physical domain, a nonstaggered mesh is used for the grid. A Chebyshev-collocation method is used normal to the wall; finite difference and compact difference methods are used in the streamwise direction; and a Fourier series is used in the spanwise direction. For time stepping, implicit Crank-Nicolson and explicit Runge-Kutta schemes are used to the time-splitting method. The influence-matrix technique is used to solve the pressure equation. At the outflow boundary, the buffer-domain technique is used to prevent convective wave reflection or upstream propagation of information from the boundary. Results of the DNS are compared with those from both linear stability theory (LST) and parabolized stability equation (PSE) theory. Computed disturbance amplitudes and phases are in very good agreement with those of LST (for small inflow disturbance amplitudes). A measure of the sensitivity of the inflow condition is demonstrated with both LST and PSE theory used to approximate inflows. Although the DNS numerics are very different than those of PSE theory, the results are in good agreement. A small discrepancy in the results that does occur is likely a result of the variation in PSE boundary condition treatment in the far field. Finally, a small-amplitude wave triad is forced at the inflow, and simulation results are compared with those of LST. Again, very good agreement is found between DNS and LST results for the 3-D simulations, the implication being that the disturbance amplitudes are sufficiently small that nonlinear interactions are negligible.

Joslin, Ronald D.↗

NASA NTRS1993

Multigrid methods and high order finite difference for flow in transition - Effects of isolated and distributed roughness elements

The high order finite difference and multigrid methods have been successfully applied to direct numerical simulation (DNS) for flow transition in 3D channels and 3D boundary layers with 2D and 3D isolated and distributed roughness in a curvilinear coordinate system. A fourth-order finite difference technique on stretched and staggered grids, a fully-implicit time marching scheme, a semicoarsening multigrid method associated with line distributive relaxation scheme, and a new treatment of the outflow boundary condition, which needs only a very short buffer domain to damp all wave reflection, are developed. These approaches make the multigrid DNS code very accurate and efficient. This makes us not only able to do spatial DNS for the 3D channel and flat plate at low computational costs, but also able to do spatial DNS for transition in the 3D boundary layer with 3D single and multiple roughness elements. Numerical results show good agreement with the linear stability theory, the secondary instability theory, and a number of laboratory experiments.

Liu, C.↗

NASA NTRS1993

Large eddy simulation of shock turbulence interaction

A nonconservative formulation of the energy equation (solving for internal energy) was used to perform large eddy simulations of compressible turbulence in Moin et al. due to its simplicity in implementing SGS models compared to the conservative formulation (solving for total energy). In problems with shocks in the domain, however, the total energy formulation is preferred due to its conservative nature. A conservative set of equations for the LES were derived from the nonconservative equations derived by Moin et al. Performance of the conservative formulation was compared with the experiment on decaying grid-generated turbulence as well as with the filtered DNS field. Various shock-capturing schemes were tested, and an ENO shock-capturing scheme of Shu and Osher was chosen for the simulation of shock/turbulence interaction. The scheme was tested and validated against the data base generated by DNS of weak shock waves. The results obtained with the essentially nonoscillatory (ENO) scheme were within 5 percent from the DNS results, and used less than 25 percent of the CPU time used in the DNS.

Lee, Sangsan↗