Features of far-downstream asymptotic velocity fluctuations in a round jet: A one-dimensional turbulence study

A turbulent round jet emerging from a circular nozzle into an infinite environment is a well-studied canonical problem that serves as a testbed for fundamental research and benchmark case for model development and engineering applications. Latter includes applications in turbulent mixing, propulsion, and reacting flows in which both the flow field (like velocity statistics and turbulent fluxes) and the physical consequences (like heat release rates in flames or noise emission from jets) are of interest. However, often not all of the quantities of interest can be measured directly, so high-fidelity numerical simulations are increasingly used to support experimental ground truth with additional insight. Furthermore, accurate and economical reduced-order models are developed based on the gained fundamental understanding in order to improve predictions for engineering applications. Latter remains a challenge as models are usually developed in or for a specific flow regime such that regime independent models are rare. New modeling strategies try to alleviate this burden, for example, by machine-learning¹ or stochastic approaches. Here, a stochastic approach is used, which is based on the one-dimensional turbulence (ODT) model² as detailed below.

Robust numerical tools must be reasonably accurate and suitable for the intended application, which is confirmed by validating model predictions against experimental data. Many experimental data are available for round jets, including mean velocity profiles and second- and third-order statistical moments of velocity fluctuations that have been obtained using hot-wire and laser-Doppler anemometry (e.g., Refs. 3–6). A round constant-temperature jet was studied by Wygnanski and Fiedler,³ whereas Hussein et al.⁴ looked at an exhausting jet system. The turbulent round jet of air was analyzed by Panchapakesan and Lumley.⁵ Xu and Antonia⁶ measured the velocity of two jets with varying inflow conditions.

In addition to experiments, there are three major types of numerical simulation techniques used in computational fluid dynamics (CFD): Reynolds-averaged Navier–Stokes simulation (RANS), large-eddy simulation (LES), and direct numerical simulation (DNS), which are briefly summarized next in the context of turbulent jets. DNS is a powerful method for studying the detailed dynamics of turbulence by directly solving the Navier–Stokes equations without the need for turbulence closure models. However, the computing effort is usually high such that the usage of DNS is limited to a relatively small fraction of the relevant parameter space, which is often insufficient for engineering applications but has led to significant fundamental insight into jets. There are various studies that dealt with utilizing DNS to simulate a spatially developing jet. Boersma et al.^7,8 simulated a low Reynolds number jet using incompressible and compressible Navier–Stokes equations and compared their DNS results with experiments. Uzun et al.⁹ performed DNS of three-dimensional subsonic turbulent round jets using fully compressible Navier–Stokes equations, which used high-order schemes together with high-order filters. Another important work is DNSs of a turbulent jet for a Reynolds number based on a nozzle exit diameter $R e D = ρ U D / μ = 2000$ jet at acoustic Mach number $M a = U / a = 1.92$⁠, where ρ is the mass density of the fluid, U is the bulk jet exit velocity, μ is the dynamic viscosity, and a is the speed of sound, by Freund et al.¹⁰ They later performed DNSs of almost sonic turbulent jets with Ma = 0.9¹¹ (based on the experimental work of Stromberg et al.¹²). Picano and Casciola¹³ performed a DNS for two axisymmetric free jets with different inlet geometry to investigate whether the asymptotic jet can be universal. Ahlman et al.¹⁴ simulate a turbulent planar wall jet that is the mixing of a passive scalar. They used compact finite difference schemes and filtering techniques. Stanley et al.¹⁵ and Reichert and Biringen¹⁶ used DNS for a compressible aircraft jet that exits into a parallel flow. Balarac et al.¹⁷ performed DNS of coaxial jets at moderate Reynolds with a high-velocity ratio.

In addition to all these DNS studies, the restriction of DNS to mildly turbulent flows at low-to-moderate Reynolds numbers often renders LES the tool of choice for moderate-to-high Reynolds number jets as relevant in applications. However, its accuracy and feasibility depend inversely on the resolution (filter length scale). It is established that a traditional LES approach should resolve more than 80% of the turbulent kinetic energy. In contrast, the unresolved 20% small-scale fluctuations are modeled using subgrid-scale (SGS) diffusive gradient-flux closure assumptions similar to what was proposed by the pioneering work of Smagorinsky.¹⁸ It is essential to mention here that in contrast to the modeling assumption of “dynamically passive” small scales in hydrodynamic turbulence, resolved small-scale turbulence is crucial for aeroacoustic applications. Small-scale mixing noise is considered the dominant noise source component in subsonic (Ma < 1) jets because of the absence of Mach wave radiation.¹⁹ Morris et al.²⁰ investigated high-speed jets with constant acoustic Mach number Ma = 1.5 at heated and unheated conditions using LES and the Ffowcs-Williams and Hawkings (FW–H) method²¹ and observed the influence of inlet conditions on the radiated sound. In a series of papers, Bogey and Bailly^22–24 systematically investigated the role of inflow conditions and Mach number and Reynolds number effects. The Re_D considered were ranging from $1.3 × 10 4$ to $3.36 × 10 5$⁠. Andersson et al.²⁵ investigated an unheated Ma = 0.75 jet with the nozzle geometry included. Shur et al.^26,27 considered off-design jets (pressure mismatched supersonic jets) and jets with synthetic chevrons. Bodony and Lele²⁸ investigated heated and unheated subsonic and supersonic jets. Uzun et al.²⁹ investigated an unheated jet at two moderate-to-high Reynolds numbers keeping Ma = 0.9 fixed. Bres and Lele,³⁰ in their recent review of jet noise, state the estimation of the missing noise from the unresolved scales as one of the remaining challenges and upcoming opportunities in the modeling of jet noise. A summary of the relevant references on the application of LES for subsonic high-Re_D turbulent round jets is given in Table I.

The map-based, stochastic, one-dimensional turbulence (ODT) model gives an alternative modeling approach. In ODT, numerical efficiency is obtained by a dimensionally reduced formulation. Fidelity, however, is addressed by resolving all relevant scales of the flow along a physical coordinate that spans the diameter of the jet and is advected downstream during a simulation run. Compared to Reynolds-averaged Navier–Stokes (RANS) or LES modeling, there is no need for closure and, thus, no eddy viscosity or turbulent Prandtl number is involved. The reduced-order formulation of the ODT model achieves significant cost reductions compared to DNS and LES through reduced spatial dimensionality while resolving spatiotemporal velocity fluctuations in contrast to RANS and, partly, LES. Hence, ODT makes small-scale resolving numerical simulations of turbulent mixing and associated noise sources feasible for the far-field of the nozzle of high Reynolds number turbulent jets. It would be appropriate to say that the current state of CFD and computational aeroacoustics (CAA) has matured enough to allow the development of reduced order or more concise models that explain which part of the flow generates the sound. However, this is insufficient for quantitative predictions across applications and is clearly limited by the missing noise due to LES filtering, particularly far downstream of the nozzle, where large domain sizes render small-scale resolution not feasible. This shortcoming is addressed by stochastic reduced-order models, such as ODT, which might provide a fast yet accurate means for engineers to evaluate the far downstream noise of new nozzle designs by propagating the immediate near–field of the nozzle from a high-resolution LES or DNS. The current approach complements the state-of-the-art DNS, LES, and RANS as it aims to represent small-scale sources with high fidelity in the far-field of the nozzle.

The present work models a subsonic unheated (constant-property) round jet using ODT as a self-contained flow model. The aim is to investigate the downstream and Reynolds number dependence of velocity fluctuations, primarily of the small-scale turbulence. The proposed methodology finds its novelty in modeling small-scale turbulence, which has been considered a critical noise generating mechanism in subsonic jets. Therefore, it needs to be addressed for the state-of-the-art aeroacoustic applications.

The paper is organized as follows. In Sec. II, a theoretical background of ODT is presented, followed by the mathematical model and simulation setup of the round jet problem in Sec. III. In Sec. IV, the results and validation are discussed. Conclusions are drawn in Sec. V.

This section summarizes the key features of the ODT model. We present the governing equations, the formulation of the eddy events, the selection of these events, and the relevant physical model parameters, starting with an overview.

One of the most significant advantages of ODT is that it allows for cost-effective simulations of high Reynolds number turbulence across the entire spectrum of dynamically applicable length scales, allowing for a physics-based representation of interactions between turbulent advection and microphysical processes. Another advantage of the model is that it exploits a degree of universality by modeling overturning fluid motions by a prescribed mapping and kernel application as detailed below.

The ODT formulation for planar configurations was introduced by Kerstein² in 1999, which was later modified to model also the effects of pressure-velocity couplings,⁴² variable density effects,⁴³ mesh adaptivity,⁵⁴ and cylindrical geometry.⁵⁵ The model was gradually expanded, validated, and applied to a number of flows, as summarized in Table II. However, the effects of Reynolds number variations for non-reactive turbulent round jets have not been investigated in detail. This is done here with an eye on turbulent velocity fluctuations to evaluate the regime-spanning modeling capabilities of ODT and to infer its suitability for aeroacoustic noise source estimation.

In the ODT model, a stochastic sampling of discrete mapping (“eddy”) is used to model the effects of nonlinear advection and fluctuating pressure forces for a notional line-of-sight through the turbulent flow. These discrete eddy events punctuate the resolved deterministic processes, which are here given by molecular diffusion and downstream advection processes. The ODT governing equations can be expressed in a temporal formulation (T-ODT) and a spatial formulation (S-ODT), briefly explained below. Even though, for the present study, S-ODT is employed due to a physically more accurate representation of local advection time scales in the flow, we begin with the T-ODT formulation to introduce the modeling concept. In T-ODT, the ODT domain is advected with a uniform bulk velocity^46,57 whereas, in S-ODT, this is done on a local basis for Langrangian grid cells.⁵⁴ Lagrangian grid cells involve a fully adaptive but conservative model formulation based on a finite-volume discretization. The application to planar geometry is detailed in Ref. 54 and to cylindrical geometry in Ref. 55. Skipping over technical details, we defer the reader to Ref. 58 for a complete derivation of the ODT equations in Lagrangian and Eulerian form, respectively.

Discrete eddy events denoted as $E ( u → )$ in Eqs. (1) and (2) aim to represent the effects of turbulent stirring motions along the ODT domain. This is done by a permutation of fluid elements using a mapping operation.² This mapping is measure-preserving in order to address mass, momentum, and energy conservation principles. It is also continuous and modifies property profiles in a scale-local fashion in order to represent turbulent cascading phenomenology in a dimensionally reduced setting. The mapping prescribes the microstructure of turbulent motions but is applied at different locations across the whole range of resolved length scales.

Different mapping functions have been developed, but the triplet map² is conventionally used as it satisfactorily captures the relevant flow physics that can be economically implemented.

Here, the so-called pseudo-planar triplet map B (PTMB) is utilized, which features an equal partitioning of the eddy size interval based on eddy size and an evaluation of the eddy rate expression using a planar analytic mapping kernel.⁵⁵ PTMB effectively suppresses sampling artifacts at the axis that would otherwise be sensible in the unmodified triplet map B (TMB), which utilizes a numerical mapping kernel for the radial coordinate in the eddy rate expression, and triplet map A (TMA), which is based on an equal-volume partitioning.⁵⁵ TMA yields relatively large near-axis modeling errors but can be expressed analytically,⁵⁹ which might be favorable for non-adaptive grids with application to pipe flow boundary layers so that near-axis properties are less of concern. For the round jet, it was demonstrated previously⁵⁵ that PTMB is able to reduce, but not entirely remove, artificially enhanced root mean square (r.m.s.) fluctuation velocities $u c , rms , i ( x ) = ⟨ u i ′ 2 ⟩ ( r = 0 , x )$⁠, i = 1, 2, and 3, at the jet centerline (“c”) at the expense of a somewhat increased model complexity. PTMB yields reasonable model fidelity on and off the jet axis with acceptable overhead in model complexity compared to TMB alone, so we utilize PTMB unless otherwise noted.

The eddy streamwise increment ξ and the mean sampling streamwise increment $ξ s$ are compared to obtain the average acceptance probability $p a = ξ s / ξ ≪ 1$ of a physically plausible eddy event. $ξ s ≪ ξ$ reflects the oversampling due to probabilistic acceptance of eddies. In this process, eddy events are assumed to be independent of each other so that the streamwise increment between two such events can be sampled economically from an exponential distribution in the framework of marked Poisson processes.⁶¹ 

Suppressing unphysically large-eddy events, which can infrequently occur during the sampling process, is often necessary. This is, in particular, the case for jets, in which the dispersion (spreading) depends on an accurate representation of turbulence properties at and around the turbulent/nonturbulent interface region. For this purpose, a large-eddy suppression (LS) mechanism is sometimes used.⁵⁷ Simple suppression based on the fraction of the domain length may be adequate for confined flows such as channel and pipe flows.^55,62 However, elapsed time (T-ODT) or, analogously, a streamwise extent (S-ODT) approach is preferred for free shear flows, such as jets.⁴⁶ Only eddy events satisfying $ξ ≤ β LS x$ for S-ODT and $τ ≤ β LS t$ for T-ODT are allowed, where $β LS$ is the large-scale suppression parameter. This model parameter limits the mean jet spreading (apex half angle) by physically constraining the realizable large scales. That is, the largest permissible eddy size l in the sampling procedure cannot exceed the current downstream distance x, or, more precisely, $x − x 0$⁠, where x₀ is the virtual origin, which is $O ( r 0 )$ and unimportant for far downstream distances $x ≫ r 0$⁠. This large-scale suppression method has frequently been used for ODT applications to planar and round jets.^46,55,63 The involved model calibrations aimed at finding a set of model parameters that applies to turbulent jets at arbitrary, but sufficiently high Reynolds number based on jet similarity arguments.

In this section, we first describe the reference case. After that, we detail the model setup by addressing the inflow condition. Last, we justify the model calibration by a sensitivity analysis to model parameter variations. These considerations are relevant for the interpretation of the results.

The aim of the present study is to investigate small-scale turbulence in the far-field of a nozzle for a high Reynolds number using ODT as stand-alone model. The reference case is a jet of air into ambient air.

Using the adopted modeling strategy, we strive to represent small-scale turbulent noise sources with high fidelity. We select the unheated round jet of Bogey and Bailley^24,32 as a reference case for model validation with respect to momentum transport and downstream evolution of turbulence properties. The nozzle diameter is $D = 0.02 m$ and the local flow velocity at Mach number is Ma = 0.9, which corresponds to a jet Reynolds number of $R e D = 4 × 10 5$⁠. Further details of the reference LES conducted by Bogey and Bailley^24,32 are given in Table III. It is also worth noting that in the present study, the jet is considered to be incompressible since the aim of this work is to study small-scale turbulence in the far-field, where turbulence is considered to be incompressible.^40,64–66 Bradshaw,⁶⁷ in his work on the effects of compressibility on turbulence, also emphasized that for axisymmetric jets, the velocity decreases rapidly with the flow coordinate, and the flow becomes incompressible after a few nozzle or body diameters downstream. We justify in  Appendix C the application of an incompressible (constant-property) flow model to a nominal Ma = 0.9 jet by much lower locally realized Mach numbers as a consequence of momentum mixing and jet dispersion.

Here, a round jet is modeled with uniform co-flow, but with the awareness that the actual flow would distort a uniform flow owing to ellipticity effects. It is expected that an assumed uniform co-flow would not have much effect on the far-field of the jet^68,69 unless it was so strong that it distorted the flow to the point that it was no longer really a free jet. So if it significantly affects only the near field of the jet, then the only region in which the shape of the co-flow profile is consequential is near the jet axis. This allows us to use the uniform co-flow, viewing it not as a model of the physical jet with co-flow but as a simple representation of ellipticity effects on the inlet-plane profile of the free jet.

The details of the near turbulent field behind the nozzle are less relevant than the well-developed far-field properties on which we focus here. Therefore, the initial velocity profile is not perturbed as in the reference data summarized above. The random breakup of the turbulent jet develops turbulent fluctuations such that there is no need to prescribe them here. More elaborate initial (inflow) conditions with anisotropic turbulent fluctuations can be used in case near-field properties are crucial.⁵⁷ 

The inflection points of the tanh transformation, $− D / 2$ and $D / 2$⁠, are $y c 1$ and $y c 2$⁠, respectively. The profile of the inflow longitudinal velocity normalized with the centerline velocity thus obtained is shown in Fig. 1. To better understand the effect of co-flow velocity on the evolution of the jet and the low-order statistical moments on radial and axial location, a parametric study is performed for $γ ∈ { 0.1 , 0.2 , 0.3 }$⁠.

Note that the addition of a uniform co-flow velocity, $u cof$⁠, to the velocity profile is essential as the evolution equations in S-ODT require $u ( r , x ) > 0$ everywhere. Otherwise, the parabolic advancement would fail by the “stall” of the simulation. This can be understood as follows. Adaptive grid cells would shrink in size upon viscous entrainment of fluid with zero axial velocity since axial momentum flux needs to be conserved. The reduction in radial ODT cell size (or, more precisely, the reduction of the axially oriented cross-sectional area of an ODT cell) compensates for the cell-based increase in momentum. As a consequence of shrinking cells, the CFL-limited diffusive advancement collapses. Therefore, finite axial velocities are needed for the model application. This is ensured during a simulation for co-flow velocities much smaller than the jet exit velocity since rare eddy events that would result in zero or negative axial velocities are rejected.⁵⁵ Moreover, the negative velocities are not unphysical, but the spatially evolving formalism cannot handle them because it would associate streamwise progress increments with negative time increments, which would be equivalent to negative viscosity during viscous progress.

The schematic for the ODT simulation setup is shown in Fig. 2. The stochastic ODT simulations of the turbulent jet are performed for the quasi-one-dimensional ODT domain. This infinitesimal sector spans the diameter of the cylindrical volume (dashed outline) that accommodates the jet (solid outline). As mentioned in the preceding section, this domain moves downstream with the flow. It is always aligned with and spans the entire diameter. The assignment of model parameters and their sensitivity to flow physics is discussed in Section III C. Table VI in  Appendix E tabulates the input parameters to the ODT simulations of the present case. As default, cold jet model parameters from⁵⁵ are taken into account, which were obtained by calibrating low-order velocity statistics of a canonical round jet to reference measurements⁴ for a single Re_D.

A total of 20 000 independent ODT simulations are performed, and the results are subject to ensemble statistics. Each of the spatial signals so generated is called a realization of the flow. Samples of the individual signals at a particular position y are then the sample realizations of the random variable x(y). These realizations are run in parallel on a local cluster. For each realization, the seed of the underlying pseudo-random generator used for probabilistic eddy acceptance changes while the initial inflow conditions remain the same.

In this section, the sensitivity of the results to the ODT model parameters (⁠ $α , C , Z , β LS$⁠) is discussed. To determine and convince ourselves of a reasonable selection of model parameters for the round jet, ODT results are compared to available subsonic reference LES of Bogey et al.³² and Thaker et al.³⁸ It is to be noted that Bogey et al.³² is a representative study of the targeted turbulent flow regime relevant to aeroacoustics. The parameter assignment study is carried out at $γ = 0.1$⁠. Previous research has revealed that flow characteristics are not universal but dependent on the configuration details. Different ODT model parameter selections are required for different applications such as wall-bounded flows,^2,42 mixing layers,⁴³ confined jets,⁵⁷ and free jets.^46,55 These studies show that the contained physics and the forcing mechanism utilized in the flow impact the model parameter selection for a specific flow configuration. We begin by looking at the model parameters for spatial simulations in Ref. 55 to establish the optimum parameter set for the current investigation, as it is comparable to the present study.

The transfer coefficient α, as discussed in Sec. II C, controls the turbulent energy exchange between the three velocity components. It can vary from zero to one, with 2/3 indicating a uniform energy distribution in the velocity components, so the same value is used for all cases in this study. It is worth noting that when one parameter is calibrated, the other parameters are held constant. ODT simulations are performed for $β LS ∈ { 0.0 , 2.5 , 3.5 , 5.0 } , Z ∈ { 1 , 400 , 1000 , 5000 }$⁠, and $C ∈ { 4.00 , 5.25 , 6.00 }$⁠. The radial profiles of the mean axial velocity at three downstream locations on the jet axis are used to calibrate the model parameters C, Z, and $β LS$ by comparison with the LES results.²⁴ 

The large-eddy suppression $β LS$ feature prevents a large-scale anomaly. The occurrence of large eddies can occasionally dominate the total transport, as their circulation time is longer than the total simulation time. Therefore, these eddies are unphysical artifacts of the stochastic approach and should be avoided. In the model,⁵⁴ there are several approaches to limit such eddies. The “fraction-of-domain,” “elapsed-time,” and “two-thirds” methods (as detailed in Ref. 54) can be used to reject such eddy events.

Figure 3 shows the impact of $β LS$ values on the S-ODT round jet, specifically on the radial profiles of the mean axial velocity u normalized by the local centerline value u_c at three downstream locations: $x / r 0 = 20 , x / r 0 = 50$⁠, and $x / r 0 = 100$⁠. The radial locations are normalized by the shifted self-similarity coordinate $( x − x 0 )$⁠,⁵ where x is the downstream location, and $x 0 = 8 r 0$ the virtual origin used in Bogey and Bailly²⁴ (for more details about the self-similarity scaling of jet data we refer to Boersma et al.⁷ and Lubbers et al.⁷⁰). The top row of Fig. 3 shows that the round jet is sensitive to $β LS$⁠. With no suppression of larger eddies at $β LS = 0$⁠, the velocity profiles spread unphysically, leading to a very high jet spread angle. The axial velocity in Fig. 4 also tends to have more noise and is relatively higher than the linear reference fit to the axial velocity of the reference data.²⁴ The linear decay rate of the velocity along the axis is a consequence of the global similarity of the round jet, as discussed in Burattini et al.⁷¹ On the other hand, $β LS = 5.0$ does not have much influence on the radial profiles of the velocities. However, a drop in axial velocity relative to the linear fit indicates that the optimum is between two and five. However, it is worth noticing here that for a co-flowing jet, the ODT data show clearly that the velocity varies nonlinearly with the distance from the source, which is in good agreement with the finding of Antonia et al.⁶⁹ and Chu et al.⁷² in their experimental study of co-flowing jets. Therefore, for the sensitivity analysis of Z and C, we choose $β LS = 3.5$⁠. Note that in Ref. 55, $β LS = 3.5$ was used for a low-Mach number jet, thus asserting that the parameter is sensitive to γ.

The model parameter Z aids in suppressing small-scale eddy events by scaling a viscous penalty term in the stochastic sampling procedure. This penalty term accounts for molecular friction and, thus, provides a weak formulation of the cutoff mechanism for small eddies. Using Z = 1, the model effectively suppresses eddy events smaller than the Kolmogorov scale. The initial value for Z was chosen based on the evaluation of turbulent jet.^55,73 A sensitivity analysis was then performed for two other values of Z to determine the optimal value used in this work. The ODT realizations were carried out for fixed $β LS = 3.5$ and C = 5.25.

Figure 3 shows the effect of Z values on the radial profiles of the mean axial velocity u normalized by the value of the local centerline $u c$ at three downstream locations. The profiles appear to collapse into each other, indicating that the jet configuration is less sensitive to the model parameter Z as mentioned above, considering previous studies. However, to ensure that the parameter has virtually no effect on the evolution of the jet, a sensitivity analysis is performed for the mean axial velocity along the centerline with respect to the downstream position. The middle plot in Fig. 4 shows $u 0 / u c$ vs $x / r 0$⁠; the similarity scaling gives a nominally linear profile where u falls off as $1 / x$⁠. The ODT simulation compares very well with the reference data⁷² after an initial induction period for $x / r 0 < 25$⁠. The dashed line in the diagram is the linear curve fit to the reference data. However, it is important to mention that a few tunable model parameters control an ODT simulation, and different combinations of these parameters might recreate additional features of the flow dynamics. A particular set of values of these parameters may repeatedly provide an acceptable mean velocity profile but not an ideal Reynolds stress. It is also possible that several pairs of values of these parameters accurately represent the natural flow dynamics in the mean velocity profile^45,55 and that one parameter, like Z in jets,⁵⁷ has a negligible effect on the mean state.

In contrast, pipe flow is sensitive to Z.⁵⁵ Therefore, Z = 400 was chosen for the present study. We discuss in more detail in  Appendix A how Z influences the participating scales and jet evolution.

The C parameter regulates the frequency of eddy events or the turbulence intensity. The flow is laminar for lower C values because fewer eddies are implemented. For large values of C, the inverse is true. The ODT simulations of the test case were carried out for $C ∈ { 4.0 , 5.25 , 6.0 }$ at $β LS = 3.5$ and Z = 400. The radial profiles of the mean axial velocity at three downstream locations are shown in the bottom row of Fig. 3. The flow dynamics remain unchanged for $y / ( x − x 0 ) < 0.1$ at $x / r 0 = 20$⁠. In principle, reducing C decreases the number of eddies implemented, resulting in a less turbulent flow with weaker momentum mixing and jet spreading. Correspondingly, a substantial rise in the slope of the mean velocity profile is observed along the radius (Fig. 3, bottom row) and along the axis (Fig. 4, rightmost panel) for the lowest selected value C = 4.0. The variation along the jet centerline of the r.m.s. fluctuation axial velocity $u c , rms ( x )$ as defined above is shown in Fig. 5. The flow dynamics remain unaltered until the end of the potential core, i.e., until $x / r 0 ≃ 11$⁠. Fewer eddies are implemented for small C, which reduces the level of turbulence of the flow. We observe a good match of the axial velocity fluctuations between the ODT results and the reference LES data³⁸ for all values of C. Altogether, the synopsis of all model calibration results suggests that C = 5.25 from Lignell et al.⁵⁵ could be used here, although it was calibrated for a jet with much larger Re_D. Due to the somewhat better far-downstream representation of the turbulent fluctuation velocity $u c , rms = u c , rms ( x )$ or, correspondingly, the centerline axial turbulence intensity $⟨ u ′ u ′ ⟩ ( r = 0 , x ) = u c , rms 2 ( x )$⁠, as shown in Fig. 5, we select C = 4 for the extended ODT simulations below.

The above discussion is significant for demonstrating how the ODT parameters impact the jet spreading and decay. The center panel in Fig. 4 shows that all values of Z yield a decay of the mean centerline axial velocity $u c ( x )$ as $u c ( x ) ∼ x − 1$⁠, which is expected from reference data (e.g., Chu et al.⁷²) It is not surprising that the smallest eddies do not influence the jet spreading and hence velocity decay. However, the left panel indicates that obtaining the correct velocity decay requires selecting the right value of $β LS$⁠. As observed already in Fig. 3, $β LS$ controls the spreading (and hence velocity decay) of the jet. In this sense, the ODT model is not predictive, as one must know the expected velocity decay rate in order to set the parameter $β LS$⁠. The right panel indicates that all 3 values of C have the same slope (scaling), only different intercepts. This means that running the “eddy clock” faster does not change the jet spreading and decay scaling.

Table IV lists the values of the model parameters used in the current study.

The present results are subject to ensemble statistics of 20000 independent ODT simulations.

Figure 6 shows the axial velocities normalized by the centerline velocity at different axial positions $x / r 0$⁠. The dots represent the mean velocity, and the blue lines represent the instantaneous velocities from a single ODT realization. With increasing axial distance, the axial velocity decays gradually and the jet spreads in radial direction. Figure 7 shows cross-stream axial-radial visualizations of the turbulent jet based on the implemented sequences of ODT eddy events. Here, an eddy event is represented by a finite-size radial interval that spans the eddy size interval for the selected location. Hence, an ODT flow realization consists of a sequence of stochastically selected eddy events that forms a 2D snapshot of the turbulent jet. Any such sequence is an ODT simulation result that yields a synthetic flow field, which contains small-scale information and aims to be statistically representative of the turbulent jet. The three flow realizations shown in Fig. 7 have been obtained for different values of γ. They reveal the multiscale and cascading properties of the turbulent flow and how both are qualitatively influenced by the co-flow velocity. As noted above, an increase in γ results in an increase in the potential core length of the jet and a decrease in the cross-stream jet propagation (spreading) rate and, hence, in a narrower apex half angle.

In this section, the effect of co-flow velocity on the developing round jet is investigated in order to assess the relevance of unresolved flow ellipticity effects by solving the parabolized ODT equations as surrogate model. We consider co-flow to jet centerline velocity ratios $γ = 0.1$⁠, 0.2, and 0.3. The axial profile of the excess centerline velocity^68,69,72 $( u c − u cof )$ is shown in Fig. 8(a). These profiles are normalized with their value at inflow plane, that is, $( u j − u cof )$⁠, where $u j$ is the mean centerline velocity at the nozzle orifice. The axial coordinate is normalized by the jet exit radius r₀. A relatively longer potential core with a weaker centerline velocity decay is observed for increasing co-flow velocity ratios. For $γ = 0.1$⁠, the velocity decay starts at the end of the potential core for about $x / r 0 = 10$⁠, which is in good agreement with the experimental⁷⁴ and LES³² data. In Fig. 8(b), after shifting the data to match the potential core length, it can be seen that the decay rate of the excess velocity is also affected by the presence of a co-flow. This decrease in the decay rate of the excess velocity and the weaker spreading of the jet, as shown in Fig. 9, are directly related to the fact that the presence of a co-flow leads to a weaker distortion and a further downstream merger of the conical shear layer at the axis. It is also evident from the plots that an assumed uniform co-flow does not have much effect in the jet far-field unless it were so strong as to distort the flow so much as to no longer truly represent a free jet.

Furthermore, Fig. 9 demonstrates that the ODT prediction for the centerline axial velocity at $γ = 0.1$ matches well with available reference LES.³⁸ Therefore, the same value of γ is chosen below for further investigation of the jet. Before proceeding with the statistical analysis, we discuss next how γ influences the developing turbulent jet.

The radial and axial dependence of conventional first- and second-order velocity statistics generated by ODT are examined in this section. Various ODT results are compared with each other and available reference numerical simulations and experimental measurements. The triplet map PTMB⁵⁵ has been used for the ODT simulations with the selected model parameters C = 4.0, $β LS = 3.5$⁠, and Z = 400 as listed in Table IV. C and $β LS$ were calibrated to ensure that the developing jet matches the reference data. Z was kept fixed after the initial sensitivity test since neither planar nor round jets are very sensitive to reasonable values of Z.^55,57

Here, a symmetric center cell is used having the fixed size $Δ r center = 6 Δ r min = 0.03 mm$ in order to separate it from the lower limit $Δ r min$⁠. We observed that a relatively large symmetric center cell at the jet axis might cause an unphysical lack in eddy events in the vicinity of the jet axis. Refer to  Appendix D for a detailed discussion on the influence of the center cell. The implemented eddy events as shown in Fig. 7 encompass a broad range of scales and transient turbulent fluctuations. These fluctuations overlay with the, on average, linearly dispersing (broadening) turbulent jet. The jet dispersion can be estimated as $y = ± m ( x − x 0 )$⁠, where m = 1/5 is the slope associated with an apex half-angle of $11.8 °$ and $x 0 = 8 r 0$ is the virtual origin (see, e.g., Hussein et al.⁴) Altogether, the small-scale resolution is crucial for detailed modeling, which is feasible for ODT as an economic microscale model. In addition, ODT is also capable of capturing relevant mean and large-scale features of turbulent jets providing means for efficient multi-scale turbulence simulations.

Figure 10 shows an ensemble-averaged quantitative characterization of the streamwise velocity profiles, which revealed a top-hat profile typical of free round jets in all cases. Profiles at ten axial locations, from 10r₀ to 100r₀, are shown. The exit top-hat velocity profiles diffused out progressively to a Gaussian profile within the first three radial distances. As downstream distance increases, the velocity distribution finally relaxes to bell-shaped profiles. After non-dimensionalizing the velocity with the centerline velocity u_c and the radial distance with $( x − x 0 )$⁠, all profiles after $x / r 0 > 50$ exhibit self-similarity as they collapse on a single curve. This result agrees with the data of Panchapakesan and Lumley⁵ and Bogey et al.^24,32 obtained in the self-similar region.

The phenomenology of a spatially developing turbulent jet is controlled predominantly by turbulent advection and to a lesser degree by molecular dissipation. This is reflected by the relative importance of the eddy-rate parameter C and the large-scale suppression parameter $β LS$ discussed above. The latter two model parameters govern the overall turbulence intensity and occurrences of large-eddy events, which rarely occur in the sampling procedure but affect the jet dispersion properties as can be inferred from Figs. 3 and 4 in combination with Fig. 7.

Conventional ensemble-based statistics of the fluctuating velocity field are calculated. For low-order fluctuation statistics of the axial velocity fluctuations $u ′$ (i = 1) as shown in Fig. 11(a), the axial development is in reasonable agreement for the entire axial range investigated with the experimental data of Arakeri et al.⁷⁵ as for the mean axial velocity u shown in Figs. 8 and 10 above. Similar observations have been reported by various studies.^11,76 The axial profile shown in Fig. 11(a) peaks at about $x = 14 r 0$⁠, slightly after the end of the potential core where the shear layers merge. This is consistent with the LES data for vanishing^24,32 and finite³⁸ co-flow velocity. These comparisons demonstrate that ODT well describes the jet transition. This is important for noise prediction since the end of the potential core is known to be a significant source of noise.²⁴ Experiments^4,5 have demonstrated that a turbulent jet becomes self-similar only approximately 100 radii downstream of the nozzle. Bogey et al.²⁴ studied the self-similar plateau for round jets at different Reynolds numbers and postulated that the plateau reaches its maximum value at $x = x c + 4 r 0$⁠. At this location, $u c , rms / u c = ⟨ u ′ u ′ ⟩ 1 / 2 / u c$ is 0.160 for ODT which matches well with 0.162 of the reference LES²⁴ as shown in Fig. 11(b). It is to be noted that the turbulence intensity increase along the jet axis is Re dependent. It occurs for a shorter downstream distance when Re is lower.^3–5,74 This partly explains the differences seen between the reference data shown.

An interesting phenomenon of self-similarity was also investigated in the present study. Experimental data have shown that the mean velocity and turbulent shear stress profiles become self-similar beyond a downstream distance of about ten orifice diameters. For other statistics, such as the turbulent velocity fluctuations, self-similarity is expected only after about 35 diameters. In LES studies,^7,24,32 the jet was analyzed only up to the first 45 diameters downstream of the orifice due to computational limitations. However, since this is not the case with ODT, the jet could be analyzed downstream in the region of self-preservation (⁠ $x / r 0 ≤ 100$⁠). Therefore, we investigated the self-similarity of the mean and fluctuation velocity profiles.

Radial profiles of the normalized axial and radial turbulence intensities are shown over a range of streamwise locations are shown in Fig. 12. The radial coordinate has been normalized with the jet half-width $r 50 %$⁠, which denotes the radius r at which the ensemble-averaged axial velocity $u ( r , x = fix . )$ has dropped to 50% of the ensemble-averaged centerline velocity $u c ( x = fix . )$ for a preselected downstream location x. The collapse of the r.m.s. velocity profiles in the streamwise direction shows the self-similar characteristics of the round jet with respect to its turbulence intensities $⟨ u i ′ 2 ⟩$⁠.

The radial velocity fluctuation variance $⟨ v ′ v ′ ⟩$ (i = 2) is degraded near the axis in contrast to the axial velocity fluctuation variance $⟨ u ′ u ′ ⟩$ (i = 1). This suggests filtering effects due to a spurious turbulent transport barrier that artificially suppresses ODT eddy events or turbulent fluctuations across the axis as discussed in  Appendix C. Even though the effect is minimized here by careful setup of the adaptive mesh, the numerical effect cannot be completely removed within the stand-alone application of the 1D model, which, therefore, remains influenced by the coordinate singularity at r = 0. This is supported by the artificially reduced velocity fluctuations on the axis for $x / r 0 < 60$ shown in Figs. 12(a) and 12(b).

Note that previous results obtained by Lignell et al.⁵⁵ [their Figs. 7–10, panel (c)] show the same effect and downstream behavior. Altogether, triplet map PTMB combined with a small symmetric center cell minimizes but does not entirely remove the imprint of the axis singularity on fluctuation statistics within the present stand-alone ODT application to the round jet. However, as shown by the mean velocity (Figs. 3, 4, 8, and 9) and the axial dependence of the velocity fluctuations (Fig. 5), artificial features at the jet axis remain confined to the vicinity of the axis which is deemed acceptable for the intended future application since turbulent noise sources associated with the outer shear region are reasonably represented.

With ODT, it is possible to obtain one-dimensional turbulence spectra for the model-resolved cross-stream direction. This has been worked out previously for isotropic turbulence^2,56 and a confined planar jet.⁵⁷ Below, we describe the planar and round jets calculation but apply it first to a planar reference case before moving to the round jet. This is done in order to evaluate the applicability of our approach that compensates for the spatial development of the mean state, thus extending two recent studies.^56,57

A one-dimensional turbulence spectrum for the jet is calculated from an N ensemble of momentary cross-stream profiles of the streamwise velocity $u ( n ) ( y ) , n = 1 , 2 , … , N$⁠, which are taken along a planar cross-stream coordinate y for a predefined downstream location x. We are interested in spectral shapes and relative changes upon parameter variations so that we consider normalized spectra, $E u ( 1 ) ( k 1 ) / E u ( 1 ) ( 0 )$⁠, where k₁ is the wavenumber of the one-dimensional spectra. The coordinate direction k₁ is identical with the momentary model-resolved diameter of the round jet.

We can now proceed in full analogy to a planar jet for which the one-dimensional turbulence spectra are obtained in a conventional way as described by Pope⁷⁷ and detailed for ODT in Klein et al.⁵⁷ for the central well-mixed region of a planar confined jet. Latter application, in fact, only mildly differs by a clipping of the cross-stream domain from Giddey et al.⁵⁶ who considered forced isotropic turbulence. The procedure from those references can be largely carried over to the round jet as detailed in the following.

Note that here fast Fourier transforms (FFTs)⁷⁷ are performed on blocks of data of size $N FFT = 1024$⁠. Note further that jet flows have kinetic energy distributed over a range of frequencies, making it difficult to use an input signal as an integer number of periods of all relevant frequencies. Applying an energy preserving window, such as the Hanning filter, to the data makes it periodic. Therefore, a Hanning window is applied to each block prior to applying the FFT in the present study. Additionally, Savitzky–Golay filter,⁷⁸ a weak moving-average filter, has been applied to far-downstream spectra in order to smooth out the short-term fluctuations primarily related to finite sample size and highlight the longer-term trends instead leaving the overall shapes of the low and moderate wavenumbers of interest unchanged.

In order to obtain a spectrum of adequate quality, spatial clipping of ODT line data has been adopted previously during the transient stage if a confined planar jet⁵⁷ to remove wall effects. For the free jet, clipping cannot be used in the same way as it would artificially modify the jet spectra. Nevertheless, it is helpful to assess the influence of the low-wavenumber variability and, hence, the robustness of the spectra computation. In order to test the robustness of the approach taken, we define clippings by cutoff regions selecting a finite ODT-line interval $y ∈ [ − Δ / 2 , Δ / 2 ]$ instead of $[ − ∞ , + ∞ ]$ in Eq. (9). In that way, Δ yields a predefined low-wavenumber cutoff in addition to the high-wavenumber cutoff that is governed by grid resolution governed by the adaptive range defined by the minimum (⁠ $Δ r min$⁠) and, possibly, the maximum (⁠ $Δ r max$⁠) mesh cell size, respectively. A sensitivity analysis of the energy spectra is performed for fixed $x / r 0$ using four different cutoff regions (CRs) $Δ n = 2 R + 2 ( n − 3 ) r 0$⁠, where $n = 1 , 2 , 3 , 4$ is an integer labeling the CR and affecting its size. R is defined by a physical measure of the jet width, so that $R ≃ r 50 %$⁠. The cutoff regions are marked by different colors in Fig. 13, where $C R 3$ corresponds to the cutoff region due to jet spreading and is given by $Δ 3 = 2 R$ for fixed downstream location x₀. Correspondingly, and in accordance with Fig. 13, we define $C R 4$ by $Δ 4 = Δ 3 + 2 r 0 , C R 2$ by $Δ 2 = Δ 3 − 2 r 0$⁠, and $C R 1$ by $Δ 1 = Δ 3 − 4 r 0$⁠. The influence of a finite cutoff region is a marginal shift in amplitude of the low-frequency component as the selected region it encloses a substantial fraction of the jet. This demonstrates the robustness of the jet spectra computations so that it is safe to let $Δ n ≃ 2 R → ∞$ independent of n.

In order to gain further confidence in the chosen approach⁵⁷ for the spectral analysis of the fluctuating velocity signals, the sensitivity analysis is performed on the planar jet of Namer and Ötügen⁷⁶ at Re_D = 7000.

Figure 14 shows the spectral distributions of velocity fluctuations at an axial location 40 nozzle radii downstream from the jet exit. $E u ( 1 ) ( k 1 )$ is normalized by the constant term $E u ( 1 ) ( 0 )$⁠. The spectrum based on $C R 3$ agrees relatively better than the other three. $C R 1$ and $C R 2$ capture lower energy levels, which can be explained by the fact that the limiting region in these two domains lies within the shear layer. $C R 4$⁠, on the other hand, exhibits higher energy at lower wavenumbers but further downstream collapses with the other spectra at high wavenumbers. A possible reason for this behavior could be the presence of lower energy eddies just outside the shear layer. Bashir and Uberoi⁸⁰ observed a similar trend in their velocity spectra using hot wires of a plane jet at Re_D = 7000. We discuss a similar observation for the test case of round jet^24,32,38 at $R e D = 4 × 10 5$ in  Appendix B.

The dependence of the one-dimensional spectrum of the streamwise velocity fluctuations on the ODT small-scale suppression parameter Z, the jet Reynolds number, and downstream location is investigated in the following. Figure 15 shows the radial turbulence spectra of the streamwise velocity fluctuations for $R e D = 4 × 10 5$ at $x / r 0 = 20$ for Z = 1 and 400 in order to analyze how fluctuation scales are affected by small-scale suppression. We recall that Z = 1 only suppresses eddy events smaller than the viscous length scale, whereas Z = 400 is typical for wall-bounded flows where it suppresses shear production in a region adjoining a rigid domain boundary, so that the law of the wall is established.² As mentioned above, $Z$ represents a local critical shear Reynolds number. Values for such a Reynolds number depend on the application, but values reported in the literature (e.g., Pope⁷⁷) and compiled for different flows are usually of the order O(20) and well below O(100). It is, therefore, physically reasonable to consider $Z ≲ 20$ and, correspondingly, $Z ≲ 400$⁠.

Figure 16 shows radial turbulence spectra $E u ( 1 ) ( k 1 )$ for fixed axial location $x / r 0 = 40$⁠. These spectra are normalized by the constant contribution $E u ( 1 ) ( 0 )$ at wavenumber zero since we are primarily interested in the relative change of the spectral shapes and lesser in absolute values. Spectra for different Reynolds numbers are shown at downstream locations $x / r 0 = 20$ and $x / r 0 = 50$ in in Figs. 16(a) and 16(b), respectively. Fairly similar spectral shapes can be discerned exhibiting the same trends for all Reynolds numbers and axial locations investigated. Latter is associated with qualitative spectral variability, whereas the former modulates the level. Kolmogorov's⁷⁹  $E u ( 1 ) ∼ k 1 − 5 / 3$ power-law scaling for the inertial range of homogeneous isotropic turbulence is only approximately realized, suggesting that the jet exhibits notable local imbalances.

Inertial range scalings can be discerned for all axial locations for the present Re_D, while a dissipation range-like scaling (for higher wavenumbers than that associated with the Kolmogorov scale) can only be discerned closer to the nozzle, where the lower, sub-mixing transition at around $x / r 0 = 20$ occurs. The behavior is in good agreement with reference data from Fellouah et al.⁸¹ Furthermore, at a larger Reynolds number, the relative contribution of the small scales is reduced in comparison with the large scales. These effects of the Reynolds number were also reported by Namer and Ötügen⁷⁶ on velocity spectra in planar jets.

Figure 17 shows normalized radial velocity spectra at different axial locations $x / r 0 ∈ [ 20 , 100 ]$ downstream of the nozzle. The inset shows the previously adopted spectra normalization without filtering of the data, whereas the main axes shows spectra normalized based on jet similarity considerations using smoothed data. Overall, the same general features as discussed in Secs. IV F and IV G can be discerned so that we focus on the spectral variability with increasing downstream distance.

In the inset, no collapse of the data is observed and finite sample variability yields statistically meaningful results only for wavenumbers k₁ with TKE larger than $E u ( 1 ) ( k 1 ) / E u ( 1 ) ( 0 ) ≃ 10 − 6$⁠, which is, hence, considered as the level of the relative numerical error of the present results. For high wavenumbers, $k 1 r 0 ≫ 1$⁠, dissipation range scaling with p = – 7 would be expected,⁷⁷ but this is not established by the present stand-alone ODT results for $x / r 0 > 20$⁠. The noisy spectral tails are approximately described by $E u ( 1 ) ( k 1 ) ∼ k 1 − 1$⁠. This is not physically significant as it indicates limitations by numerical interpolation, random noise, and filtering effects. It has, in fact, been shown previously that stand-alone ODT does not reproduce the expected dissipation range exponent,^2,56,57 which, therefore, constitutes a modeling error on the sub-Kolmogorov scales. However, the key point is that ODT is able to capture the spectral evolution of spatial jet to a level of fidelity that is comparable to experiments^24,76,82 but can neither be matched by averaged-equation (RANS) solvers and coarse-resolution filtering (LES) approaches.

In order to rule spurious effects due to modeling artifacts, we performed additional planar jet ODT simulations based on the planar setup described in Kein et al.⁵⁷ but now applied to reference data of Namer and Otügen.⁷⁶ The results of these additional simulations are shown in Fig. 14. It is interesting to observe an approximate horizontal collapse of the radial spectra in the inertial region for most of the curves. This suggests that eddies of the same size contain the same amount of relative energy at different axial locations in the flow, which is consistent with the results of Namer and Otügen.⁷⁶ 

Coming back to the round jet spectra in the main panel of Fig. 17, we first note that the noisy spectral tails have been clipped and spectra have been smoothed as described above for visual clarity. The spatial dependence of the fluctuations is encoded in the jet spectra, so spectra can only be self-similar if the jet is self-similar. Based on linear jet spreading, it is reasonable to assume a transformation based on large-scale jet similarity that will at least approximately collapse the jet spectra downstream of the nozzle. For fixed x, spatial distribution of the streamwise fluctuation velocity $u rms ( y )$ over the planar cross-stream coordinate y suggests notable low wavenumber content close to the cutoff wavenumber $k 1 , l o w ∼ r 50 % − 1$⁠. Hence, the resolved range of spatial scales for selected axial location x should scale with $k 1 , l o w$⁠. We correspondingly normalize the wavenumber as $k 1 r 50 %$ in order to shift all spectra horizontally to common range. Next, the streamwise dependence of the jet spectra manifests itself in the streamwise decrease in the magnitude of $E u ( 1 ) ( k 1 )$⁠. Normalization with $E u ( 1 ) ( 0 )$ obscures this physical effect. Therefore, for an the assessment of the far-downstream spectral similarity, it is more appropriate to scale $E u ( 1 ) ( k 1 )$ by the spectral bulk fluctuation kinetic energy density that is given by the product of the squared centerline fluctuation velocity $u c , rms 2$ and the half-width of the jet $r 50 %$ along with a normalization of the spectral k₁ coordinate by taking $r 50 %$ as the relevant length scale. The scale velocity for the turbulence kinetic energy is hence related to the turbulence intensity rather than the ensemble-averaged (mean) velocity of the jet. We assume that all global measures of the bulk fluctuation velocity (like the maximum, mean, or centerline value) are proportional to each other so that the centerline value $u c , rms = u rms ( y = 0 )$ for any selected x is an appropriate choice.

As demonstrated in Fig. 17, the downstream jet spectra collapse to some extend indicating approximate validity of the assertion that the flow is governed by the large-scales. There are differences that prevent the approach from being perfect. In addition to modeling errors, in general it is also the downstream decrease in turbulence intensity that manifests itself by local finite Re effects for $x / r 0 > 80$ but transient effects for $x / r 0 < 40$⁠, so that the spectral collapse is best for $x / r 0 ∈ [ 40 , 80 ]$ for the present Re_D investigated.

The ODT results shown in Fig. 17 furthermore suggest that several local effective scalings of the form $E u ( k 1 ) ∼ k 1 p$ are present. These scalings are tangential to the spectra but there is no unique p that would describe the spectra well across a finite range of wavenumbers. For locally homogeneous isotropic turbulence, that may be realized for high asymptotic Re_D, an inertial range with $p ≈ − 5 / 3$ may be expected based on the K41 theory.⁷⁹ This expectation is only approximately met in the range $4 ≲ k 1 r 50 % ≲ 8$ for the jet spectrum at $x / r 0 = 20$⁠, which is the location closest to the nozzle so that it indicates a transient phenomenon in the turbulent jet investigated. The model results suggest that large-scale jet similarity is approximately valid but requires very high Reynolds number to yield far-downstream turbulence of high enough intensity to become universal.

The turbulent far-field of an unheated round jet with nominal Mach number Ma = 0.9 and jet Reynolds number up to $R e D = 4 × 10 5$ is investigated with the aid of the one-dimensional turbulence (ODT) model. ODT is a high-fidelity stochastic flow model that aims to resolve all relevant scales of the turbulent flow but only along a notional line-of-sight that spans the jet diameter and is advected downstream with the flow during a simulation run.

The reduced dimensional ODT formulation makes it feasible to resolve small-scale turbulent fluctuations up to axial locations $x ∼ 100 r 0$ downstream of the nozzle with radius r₀. Such large axial extents are challenging for high-resolution LES and DNS due to the associated resolution requirements and computational costs required for gathering flow statistics. ODT results suggest that statistical jet similarity is approximately achieved for $x / r 0 ≳ 60$⁠, which is in agreement with reference experiments.⁷⁵ Nevertheless, there is a demand for advanced numerical methods that are able to provide a detailed representation of turbulent fluctuation phenomenology down to the smallest scales of the flow. One such example is jet noise prediction and modeling for engineering applications. Therefore, the present study is a step toward quantitative estimations of turbulent and high-frequency mixing noise sources. It is intended to use the model representation of the local eddy turnover-time and length scales for turbulent noise source estimation by propagating model-resolved kinematic pressure fluctuations in the acoustic near-field to the acoustic far-field utilizing a Ffowcs Williams and Hawkings²¹ (FWH) approach. Capturing the scales and spectra of the velocity field that governs the noise sources is hence crucial for developing economic computer models of subsonic turbulent jets at high Reynolds numbers but a standing challenge for numerical models.

The present model results for the velocity field demonstrate that ODT as a stand-alone tool is able to capture some relevant flow features of round jets. This concerns the spatial dependencies of low-order statistical moments on radial and axial location, as well as their parametric dependence on the co-flow–velocity ratio and the jet Reynolds number. A reasonably accurate representation of the spatially developing, turbulent jet, particularly at far downstream distances, including turbulent fluctuations and spectra, has been demonstrated and forms the prerequisite for turbulent noise estimation. The mean flow and turbulence properties in the far-field of the nozzle are in satisfactory agreement with available reference data. The jet dispersion angle and the virtual origin are accurately reproduced, but differences can be discerned for the near field around the nozzle (⁠ $x / r 0 ≤ 20$⁠). In this region, modeling errors manifest themselves primarily by an unphysical degradation of velocity fluctuations due to the discrete mappings influenced by the polar coordinate singularity at the axis. The influence of modeling errors associated with the axis treatment in the dimensionally reduced model is, therefore, confined to a small volume fraction of the jet. The more important aspects are that the turbulence properties and thus the entrainment across the outer shear region and various radial turbulence spectra are adequately captured with the present model setup. In particular, it has been demonstrated that jet similarity is approximately governed by the large-scale fluctuations but also affected by small-scale turbulence. Small-scale fidelity is crucial for future model applications to turbulent jet noise modeling, which demands the simultaneous resolution of regions with zero and high turbulence intensity. The latter constitute the main sources of mixing noise, and the former introduces the challenge for modeling. Altogether, we have demonstrated the general suitability of ODT for small-scale resolving numerical simulations of turbulent round jets relevant for applications in jet noise prediction. In order to overcome some fundamental limitations of the dimensionally reduced model without introducing couplings, we suggest conducting DNS or high-resolution LES only for the potential-core region and propagating the turbulent far-field with ODT to make future noise predictions based on the entire jet.

We thank Juan Alí Medina Méndez for fruitful discussions and comments on the manuscript. We furthermore thank David O. Lignell for making the adaptive ODT implementation available. The public code version used to generate the results is freely available at https://github.com/BYUignite/ODT.

The present work is funded by the German research foundation (DFG) under the Walter Benjamin program, Grant No. SH 1800/1-1.

The authors have no conflicts to disclose.

Sparsh Sharma: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Marten Klein: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Writing – review and editing (equal). Heiko Schmidt: Formal analysis (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – review and editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The model parameter Z > 0 suppresses unphysically small-eddy events based on energetic considerations for the momentary flow state. Suppressed-eddy events are assumed to not contribute to the turbulent mixing while significantly reducing computational efficiency. Based on the model formulation outlined previously, it is clear that for any simulation case and any parameter selections made (other than Z), there are sufficiently high values for Z so that all eddies are suppressed. We have discussed previously how Z affects the velocity statistics. Here, we show the results of a single ODT realization for Z = 1, 400, 1000, and 5000 to visualize the effects of small-scale suppression. As expected and seen in Figs. 18(a)–18(d), less eddy events occur for higher values of Z. A drastic reduction but not a complete cutoff of the implemented eddy events is observed for unphysically large values of $Z ≳ 1000$⁠. Typical values for Z are well below 1000 also for multi-physical round jets.^55,62 The scale range of accepted eddy events is almost unaltered for these values in the round jet.

Section IV F of this article discusses the selection of the radial truncation by considering a cutoff region (CR) based on the cross-sectional diameter (radial interval) $[ − R , R ]$ for the evaluation of the Fourier integral given in Eq. (11). The analysis presented above was conducted for an experimental planar jet of Namer and Ötügen⁷⁶ at Re_D = 7000, which also helped us to validate our approach to calculate the spectra. In this section, we present the results of the same sensitivity analysis on the round jet test case^24,32,38 at $R e D = 4 × 10 5$⁠. Four different cutoff regions are used as defined above in Fig. 13. The radial turbulence spectra $E u ( 1 ) ( k 1 )$ are calculated for $x / r 0 = 20$ and $x / r 0 = 50$ across $− R < y < R$ as a function of $k 1 r 0$⁠. Spectra computed in that way are shown in Figs. 19(a) and 19(b). They are normalized by $E u ( 1 ) ( 0 )$⁠, which is the constant (wavenumber zero) contribution.

It is common to define the jet Mach (Ma) and Reynolds (Re_D) numbers based on the bulk or maximum (centerline) mean jet exit velocity as measured at the orifice of the nozzle. This serves to characterize the jet as a whole. The values of Ma and Re_D are representative for the near field with the initial breakup and the potential core, but less for the well-developed turbulent far-field in which flow velocities have significantly decayed due to momentum dispersion. Here, we consider an approximately constant-temperature, high-Re_D turbulent jet in which fluid properties and the speed of sound are approximately constant so that only variable flow velocities need to be considered.

The artifacts observed for a large center cell (Fig. 20 in the  Appendix D) are independent of the mapping function used, suggesting that small mesh sizes are required in the vicinity of the jet axis in order to yield grid-independent results. For the intended future application of turbulent jet-noise estimation, it is also important to capture dynamic details in the outer shear region containing the turbulent/nonturbulent interface, supposedly dominating sources for mixing noise. It is shown below that the flow statistics are captured everywhere and that the outer shear region, in particular, is reasonably well represented by S-ODT utilizing PTMB and a small symmetric center cell. Note that the center cell is symmetric in order to avoid flux imbalances across the coordinate singularity at the jet axis that would otherwise occur in the cell-based interpolation scheme.⁵⁵ 

Figure 20 shows a spatially evolving turbulent jet due to an ODT simulation performed with a symmetric center cell of size $Δ r center = Δ r max = 8 mm$ instead of the default $Δ r center = 6 Δ r min = 0.03 mm$⁠. The rationale behind a large symmetric center cell is increased coupling of subdomains with r < 0 and r > 0 across the coordinate singularity at r = 0 by grid-level interpolation in the discretized equations. An adverse effect of a large center cell is, however, that the mesh can only be gradually refined (2.5 rule⁵⁵) so that the near-axis region is subject to implicit filtering of small and mesoscale features. This explains the notable but unphysical lack in eddy events that can be observed close to r = 0 in Fig. 20. This over-weights the potential risk of subdomain decoupling by less molecular fluxes across r = 0 since turbulent transport is significantly more efficient and not inhibited when the grid is fine enough. Hence, care has to be taken to select a small-enough center cell that is not overly restrictive but yields grid-independent results. Nevertheless, the filtering effect remains localized so that the outer region containing the turbulent/nonturbulent interface is still reasonably represented even for the case setup with a coarse center cell.

The ODT code⁸⁶ is an open-source, object-oriented C++ implementation of ODT. Here, we provide the values of the standard variables used to conduct the simulations. These are the default values based on Lignell,⁵⁵ but we have varied them as described in the manuscript.