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# Compressibility effects on statistics and coherent structures of compressible turbulent mixing layers

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## Abstract

The effects of compressibility on the statistics and coherent structures of a temporally developing mixing layer are studied using numerical simulations at convective Mach numbers ranging from $M_c=0.2$ to $1.8$ and at Taylor Reynolds numbers up to 290. As the convective Mach number increases, the streamwise dissipation becomes more effective to suppress the turbulent kinetic energy. At $M_c=1.8$ , the streamwise dissipation increases much faster than the other two components in the transition region, even larger than pressure–strain redistribution, correlating with the streamwise elongated vortical structures at a higher level of compressibility. We confirm the existence of the large-scale high- and low-speed structures in the mixing layers, which accompany the spanwise Kelvin–Helmholtz rollers at low convective Mach number and dominate the mixing layer at higher convective Mach number. Conditional statistics demonstrate that the large-scale low-speed structures are lifted upwards by a pair of counter-rotating quasi-streamwise rollers flanking the structures. The small-scale vortical structures have an apparent preference for clustering into the top of the low-speed regions, which is directly associated with high-shearing motions on top of the low-speed structures. The high-speed structures statistically exhibit central symmetry with the low-speed structures. The statistics and dynamics of large-scale high- and low-speed structures in the compressible mixing layers resemble those in the outer region of the turbulent boundary layers, which reveals the universality of the large-scale structures in free shear and wall-bounded turbulence. A conceptual model is introduced for the large-scale high- and low-speed structures in turbulent mixing layers.

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Compressible wall-bounded turbulence is generally assumed to be devoid of genuine compressibility effects, meaning that the effect of finite fluid dilatation is regarded as “small,” at least in the absence of disturbing pressure gradients. In the present paper we attempt to answer the basic question of how small these effects are, by interrogating a DNS database of compressible channel flow and by using Helmholtz decomposition to infer the relative magnitude and correlations between the solenoidal and the dilatational parts of turbulence velocity fields. Not surprisingly, we find dilatational velocity fluctuations to be much smaller than solenoidal ones, but perhaps unexpectedly, we find that finite correlation between the two components accounts for a nonnegligible fraction (about 10%) of the turbulent shear stress near walls, and for up to 4% of the wall skin friction. Quadrant analysis of the dilatational velocity fluctuations shows that the largest contribution to the turbulent shear stress results from significant correlation between positive streamwise solenoidal velocity fluctuations (i.e., high-speed streaks), and positive vertical dilatational velocity fluctuations, which tend to mitigate the intensity of wall-ward sweep events.
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Aspects of turbulent shear-layer mixing are investigated over a range of shear-layer Reynolds numbers, $Re_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x0394}U\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D708}$ , based on the shear-layer free-stream velocity difference, $\unicode[STIX]{x0394}U$ , and mixing-zone thickness, $\unicode[STIX]{x1D6FF}$ , to probe the role of initial conditions in mixing stages and the evolution of the scalar-field probability density function (p.d.f.) and variance. Scalar transport is calculated for unity Schmidt numbers, approximating gas-phase diffusion. The study is based on direct-numerical simulation (DNS) and large-eddy simulation (LES), comparing different subgrid-scale (SGS) models for incompressible, uniform-density, temporally evolving forced shear-layer flows. Moderate-Reynolds-number DNS results help assess and validate LES SGS models in terms of scalar-spectrum and mixing estimates, as well as other metrics, to $Re_{\unicode[STIX]{x1D6FF}}\lesssim 3.3\times 10^{4}$ . High-Reynolds-number LES investigations to $Re_{\unicode[STIX]{x1D6FF}}\lesssim 5\times 10^{5}$ help identify flow parameters and conditions that influence the evolution of scalar variance and p.d.f., e.g. marching versus non-marching. Initial conditions that generate shear flows with different mixing behaviour elucidate flow characteristics in each flow regime and identify elements that induce p.d.f. transition and scalar-variance behaviour. P.d.f. transition is found to be largely insensitive to local flow parameters, such as $Re_{\unicode[STIX]{x1D6FF}}$ , or a previously proposed vortex-pairing parameter based on downstream distance, or other equivalent criteria. The present study also allows a quantitative comparison of LES SGS models in moderate- and high- $Re_{\unicode[STIX]{x1D6FF}}$ forced shear-layer flows.
Article
Cascades of temperature and entropy fluctuations are studied by numerical simulations of stationary three-dimensional compressible turbulence with a heat source. The fluctuation spectra of velocity, compressible velocity component, density and pressure exhibit the $-5/3$ scaling in an inertial range. The strong acoustic equilibrium relation between spectra of the compressible velocity component and pressure is observed. The $-5/3$ scaling behaviour is also identified for the fluctuation spectra of temperature and entropy, with the Obukhov–Corrsin constants close to that of a passive scalar spectrum. It is shown by Kovasznay decomposition that the dynamics of the temperature field is dominated by the entropic mode. The average subgrid-scale (SGS) fluxes of temperature and entropy normalized by the total dissipation rates are close to 1 in the inertial range. The cascade of temperature is dominated by the compressible mode of the velocity field, indicating that the theory of a passive scalar in incompressible turbulence is not suitable to describe the inter-scale transfer of temperature in compressible turbulence. In contrast, the cascade of entropy is dominated by the solenoidal mode of the velocity field. The different behaviours of cascades of temperature and entropy are partly explained by the geometrical properties of SGS fluxes. Moreover, the different effects of local compressibility on the SGS fluxes of temperature and entropy are investigated by conditional averaging with respect to the filtered dilatation, demonstrating that the effect of compressibility on the cascade of temperature is much stronger than on the cascade of entropy.
Article
This paper quantifies the instantaneous form of large-scale turbulent structures in canonical smooth-wall boundary layers, demonstrating that they adhere to a form that is consistent with the self-sustaining streak instability model suggested by Flores & Jiménez ( Phys. Fluids , vol. 22, 2010, 071704) and Hwang & Cossu ( Phys. Fluids , vol. 23, 2011, 061702). Our motivation for this study stems from previous observations of large-scale streaks that have been spatially locked in position within spanwise-heterogeneous boundary layers. Here, using similar tools, we demonstrate that the randomly occurring large-scale structures in canonical layers show similar behaviour. Statistically, we show that the signature of large-scale coherent structures exhibits increasing meandering behaviour with distance from the wall. At the upper edge of the boundary layer, where these structures are severely misaligned from the main-flow direction, the induced velocities associated with the strongly yawed vortex packets/clusters yield a significant spanwise-velocity component leading to an apparent oblique coherence of spanwise-velocity fluctuations. This pronounced meandering behaviour also gives rise to a dominant streamwise periodicity at a wavelength of approximately $6\unicode[STIX]{x1D6FF}$ . We further statistically show that the quasi-streamwise roll-modes formed adjacent to these very large wavy motions are often one-sided (spanwise asymmetric), in stark contrast to the counter-rotating form suggested by conventional conditionally averaged representations. To summarise, we sketch a representative picture of the typical large-scale structures based on the evidence gathered in this study.
Article
The effect of compressibility on the small scale properties of stationary homogeneous shear turbulence (HST) is studied in a rectangular domain of size 4π × 2π × 2π by numerical simulations at four turbulent Mach numbers of Mt = 0.14, 0.2, 0.4, and 0.6. Compared to the compressible homogeneous isotropic turbulence (HIT), the small scale properties of HST are more sensitive to Mach number change and the velocity derivatives show a clear deviation from isotropy. At M t = 0.14, the preferred eigenvalue ratio of the strain rate tensor is very close to −4:1:3 reported in incompressible turbulence. As M t increases, the conditional probability density functions of the normalized eigenvalues become more dependent on the dilatation, and its ratio tends to −1:0:0 in the strong compression regions at M t = 0.6, indicating the sheet-like structures of localized shock waves. The alignments between vorticity and eigenvectors are similar to the observations in HIT. After the Helmholtz decomposition, it is found that the compressible vortex stretching term is the primary cause for the enhancement of enstrophy production at high turbulent Mach numbers.
Article
We investigate the behaviour of large-scale coherent structures in a spanwise-heterogeneous turbulent boundary layer, using particle image velocimetry on multiple orthogonal planes. The statistical three-dimensionality is imposed by a herringbone riblet surface, although the key results presented here will be common to many cases of wall turbulence with embedded secondary flows in the form of mean streamwise vortices. Instantaneous velocity fields in the logarithmic layer reveal elongated low-momentum streaks located over the upwash-flow region, where their spanwise spacing is forced by the $2\unicode[STIX]{x1D6FF}$ periodicity of the herringbone pattern. These streaks largely resemble the turbulence structures that occur naturally (and randomly located) in spanwise-homogeneous smooth-/rough-wall boundary layers, although here they are directly formed by the roughness pattern. In the far outer region, the large spanwise spacing permits the streaks to aggressively meander. The mean secondary flows are the time-averaged artefact of the unsteady and spanwise asymmetric large-scale roll modes that accompany these meandering streaks. Interestingly, this meandering, or instability, gives rise to a pronounced streamwise periodicity (i.e. an alternating coherent pattern) in the spatial statistics, at wavelengths of approximately 4.5 $\unicode[STIX]{x1D6FF}$ . Overall, the observed behaviours largely resemble the streak-instability model that has been proposed for the buffer region, only here at a much larger scale and at a forced spanwise spacing. This observation further confirms recent observations that such features may occur at an entire hierarchy of scales throughout the turbulent boundary layer.
Article
The effects of Mach number on the spectra and statistics of stationary compressible homogeneous shear turbulence (HST) are studied using numerical simulations in a rectangular domain (Lx = 4π, Ly = Lz = 2π) at turbulent Mach numbers from 0.05 to 0.66 and Taylor Reynolds numbers from 40 to 100. Long-term simulation results show that a statistically stationary state is obtained and the flow meets the strong acoustic equilibrium assumption at Mt ≈ 0.4. The analysis of spectral properties indicates that velocity and pressure tend toward a Mach number scaling of Mt2 suggested by acoustic dynamics at Mt ≳ 0.3. As for one-point statistics, it is found that a Mt4 scaling predicted by pseudo-sound theory holds for normalized compressible kinetic energy, Kc/Ks, at the small turbulent Mach number Mt ≲ 0.1. Kc/Ks approaches a Mt2 scaling at relatively higher turbulent Mach numbers, which is consistent with the spectral results. The normalized compressible dissipation rate, ϵc/ϵs, is nearly independent of Taylor Reynolds number and exhibits the same Mt4 scaling at small turbulent Mach numbers. The root mean square values of pressure, density, and temperature of compressible HST show good agreement with the Mt2 scaling, with the coefficient approximately doubled as compared with the compressible isotropic turbulence.
Article
This article discusses the description of wall-bounded turbulence as a deterministic high-dimensional dynamical system of interacting coherent structures, defined as eddies with enough internal dynamics to behave relatively autonomously from any remaining incoherent part of the flow. The guiding principle is that randomness is not a property, but a methodological choice of what to ignore in the flow, and that a complete understanding of turbulence, including the possibility of control, requires that it be kept to a minimum. After briefly reviewing the underlying low-order statistics of flows at moderate Reynolds numbers, the article examines what two-point statistics imply for the decomposition of the flow into individual eddies. Intense eddies are examined next, including their temporal evolution, and shown to satisfy many of the properties required for coherence. In particular, it is shown that coherent structures larger than the Corrsin scale are a natural consequence of the shear. In wall-bounded turbulence, they can be classified into coherent dispersive waves and transient bursts. The former are found in the viscous layer near the wall, and as very large structures spanning the entire boundary layer. Although they are shear-driven, these waves have enough internal structure to maintain a uniform advection velocity. Conversely, bursts exist at all scales, are characteristic of the logarithmic layer, and interact almost linearly with the shear. While the waves require a wall to determine their length scale, the bursts are essentially independent from it. The article concludes with a brief review of our present theoretical understanding of turbulent structures, and with a list of open problems and future perspectives. ‘ Chance is the name we give to what we choose to ignore (Voltaire) ’
Article
Kinetic energy transfer in compressible isotropic turbulence is studied using numerical simulations with solenoidal forcing at turbulent Mach numbers ranging from 0.4 to 1.0 and at a Taylor Reynolds number of approximately 250. The pressure dilatation plays an important role in the local conversion between kinetic energy and internal energy, but its net contribution to the average kinetic energy transfer is negligibly small, due to the cancellation between compression and expansion work. The right tail of probability density function (PDF) of the subgrid-scale (SGS) flux of kinetic energy is found to be longer at higher turbulent Mach numbers. With an increase of the turbulent Mach number, compression motions enhance the positive SGS flux, and expansion motions enhance the negative SGS flux. Average of SGS flux conditioned on the filtered velocity divergence is studied by numerical analysis and a heuristic model. The conditional average of SGS flux is shown to be proportional to the square of filtered velocity divergence in strong compression regions for turbulent Mach numbers from 0.6 to 1.0. Moreover, the antiparallel alignment between the large-scale strain and the SGS stress is observed in strong compression regions. The inter-scale transfer of solenoidal and compressible components of kinetic energy is investigated by Helmholtz decomposition. The SGS flux of solenoidal kinetic energy is insensitive to the change of turbulent Mach number, while the SGS flux of compressible kinetic energy increases drastically as the turbulent Mach number becomes larger. The compressible mode persistently absorbs energy from the solenoidal mode through nonlinear advection. The kinetic energy of the compressible mode is transferred from large scales to small scales through the compressible SGS flux, and is dissipated by viscosity at small scales.
Article
Jets with Mach numbers $M\gtrsim 1.5$ are well known to emit an intense, fricative, so-called crackle sound, having steep compressions interspersed with weaker expansions that together yield a positive pressure skewness $S_{k}>0$ . Its shock-like features are obvious hallmarks of nonlinearity, although a full explanation of the skewness is lacking, and wave steepening alone is understood to be insufficient to describe its genesis. Direct numerical simulations of high-speed free-shear flows for Mach numbers $M=0.9$ , $1.5$ , $2.5$ and $3.5$ in the Reynolds number range $60\leqslant Re_{\unicode[STIX]{x1D6FF}_{m}}\leqslant 4200$ are used to examine the mechanisms leading to such pressure signals, especially the pressure skewness. For $M=2.5$ and $3.5$ , the pressure immediately adjacent the turbulence already has the large $S_{k}\gtrsim 0.4$ associated with jet crackle. It also has a surprisingly complex three-dimensional structure, with locally high pressures at compression-wave intersections. This structure is transient, and it simplifies as radiating waves subsequently merge through nonlinear mechanisms to form the relatively distinct and approximately two-dimensional Mach-like waves deduced from laboratory visualizations. A transport equation for $S_{k}$ is analysed to quantify factors affecting its development. The viscous dissipation that decreases $S_{k}$ is balanced by a particular nonlinear flux, which is (of course) absent in linear acoustic propagation and confirmed to be independent of the simulated Reynolds numbers. Together these effects maintain an approximately constant $S_{k}$ in the near acoustic field.
Article
Shocklet statistics in compressible isotropic turbulence are studied by using numerical simulations with solenoidal forcing, at the turbulent Mach number Mt ranging from 0.5 up to 1.0 and at the Taylor Reynolds number Reλ ranging from 110 to 250. A power-law region of the probability density function (PDF) of the shocklet strength Mn−1 (Mn is the normal shock Mach number) is observed. The magnitude of the power-law exponent is found to decrease with the increase of Mt. We show that the most probable shocklet strength is proportional to Mt3, and the shocklet thickness corresponding to the most probable shock Mach number is proportional to Mt−2 in our numerical simulations. The PDFs of the jumps of the velocity and thermodynamic variables across a shocklet exhibit a similar power-law scaling. The statistics of the jumps of the velocity and thermodynamic variables are further investigated by conditioned average. Nonlinear models for the conditional average of the jumps of the velocity and thermodynamic variables are developed and verified.
Article
The production of turbulent kinetic energy (TKE) and flow structures in compressible homogeneous turbulent shear flow (HTSF) are investigated by using direct numerical simulation. A theoretical analysis suggests that the vertical turbulent transport process should be responsible for the production of TKE in HTSF. It is manifested based on a conditional average method that the pure TKE production becomes increasingly larger in strain regions than in vortex regions of the flow. The velocity-derivative correlation in the shear plane is employed to identify the streaky structures in HTSF, which also tend to occur predominantly in strain regions of turbulence. Localized analyses of conditional averages reveal that the streaky structures in compressible HTSF are closely related to the negative productions of TKE. A comparative study implies that flow compressibility has a considerable effect on the spatial distributions and patterns of the strain- and vortex-dominated fields, which in turn cause the discrepancies in distribution of the TKE production and streaky structures between incompressible and compressible HTSFs.
Article
Direct numerical simulations (DNS) of temporally evolving shear layers have been performed to study the entrainment of irrotational flow into the turbulent region across the turbulent/non-turbulent interface (TNTI). Four cases with convective Mach number from 0.2 to 1.8 are used. Entrainment is studied via two mechanisms; nibbling, considered as vorticity diffusion across the TNTI, and engulfment, the drawing of the pockets of the outside irrotational fluid into the turbulent region. The mass flow rate due to nibbling is calculated as the product of the entrained mass flux with the surface area of the TNTI. It is found that increasing the convective Mach number results in a decrease of the average entrained mass flux and the surface area of the TNTI. For the incompressible shear layer the local entrained mass flux is shown to be highly correlated with the viscous terms. However, as the convective Mach number increases, the mass fluxes due to the baroclinic and the dilatation terms play a more important role in the local entrainment process. It is observed that the entrained mass flux is dependent on the local dilatation and geometrical shape of the TNTI. For a compressible shear layer, most of the entrainment of the irrotational flow into the turbulent region due to nibbling is associated with the compressed regions on the TNTI. As the convective Mach number increases, the percentage of the compressed regions on the TNTI decreases, resulting in a reduction of the average entrained mass flux. It is also shown that the local shape of the interface, looking from the turbulent region, is dominated by concave shaped surfaces with radii of curvature of the order of the Taylor length scale. The average entrained mass flux is found to be larger on highly curved concave shaped surfaces regardless of the level of dilatation. The mass fluxes due to vortex stretching, baroclinic torque and the shear stress/density gradient terms are weak functions of the local curvatures on the TNTI, whereas the mass fluxes due to dilatation and viscous diffusion plus the viscous dissipation terms have a stronger dependency on the local curvatures. As the convective Mach number increases, the probability of finding highly curved concave shaped surfaces on the TNTI decreases, whereas the probability of finding flatter concave and convex shaped surfaces increases. This results in a decrease of the average entrained mass flux across the TNTI. Similar to the previous works on jets, the results show that the contribution of the engulfment to the total entrainment is small for both incompressible and compressible mixing layers. It is also shown that increasing the convective Mach number decreases the velocities associated with the entrainment, i.e. induced velocity, boundary entrainment velocity and local entrainment velocity.
Article
The compressibility effects on the structural evolution of the transitional high-speed planar wake are studied. The relative Mach number ( $Ma_{r}$ ) of the laminar base flow modifies two fundamental features of planar wake transition: (i) the characteristic length scale defined by the most unstable linear mode; and (ii) the domain of influence of the structures within the staggered two-dimensional vortex array. Linear stability results reveal a reduced growth (approximately 30 % reduction up to $Ma_{r}=2.0$ ) and a quasilinear increase of the wavelength of the most unstable, two-dimensional instability mode (approximately 20 % longer over the same $Ma_{r}$ range) with increasing $Ma$ . The primary wavelength defines the length scale imposed on the emerging transitional structures; naturally, a longer wavelength results in rollers with a greater streamwise separation and hence also larger circulation. A reduction of the growth rate and an increase of the principal wavelength results in a greater ellipticity of the emerging rollers. Compressibility effects also modify the domain of influence of the transitional structures through an increased cross-wake and inhibited streamwise communication as characteristic paths between rollers are deflected due to local $Ma$ gradients. The reduced streamwise domain of influence impedes roller pairing and, for a sufficiently large relative $Ma$ , pairing is completely suppressed. Thus, we observe an increased two-dimensionality with increasing Mach number: directly contrasting the increasing three-dimensional effects in high-speed mixing layers. Temporally evolving direct numerical simulations conducted at $Ma=0.8$ and 2.0, for Reynolds numbers up to 3000, support the physical insight gained from linear stability and vortex dynamics studies.
Article
Quadrant analysis is a simple, but quite useful, turbulence data-processing technique that has been widely used, principally in the investigation of turbulent shear flows. This article traces the origins of the technique and reviews how it has been applied during the more than 40 years since it was conceived. Applications are highlighted that have expanded the technique beyond its original formulation.
Article
The local flow topology is studied using the invariants of the velocity gradient tensor in compressible turbulent mixing layer via direct numerical simulation (DNS) data. The topological and dissipating behaviours of the flow are analysed in two different regions: in proximity of the turbulent/non-turbulent interface (TNTI), and inside the turbulent region. It is found that the distribution of various flow topologies in regions close to the TNTI differs from inside the turbulent region, and in these regions the most probable topologies are non-focal. In order to better understand the behaviour of different flow topologies, the probability distributions of vorticity norm, dissipation and rate of stretching are analysed in incompressible, compressed and expanded regions. It is found that the structures undergoing compression-expansion in axial-radial directions have the highest contraction rate in locally compressed regions, and in locally expanded regions the structures undergoing expansion-compression in axial-radial directions have the highest stretching rate. The occurrence probability of different flow topologies conditioned by the dilatation level is presented and it is shown that the structures in the locally compressed regions tend to have stable topologies while in locally expanded regions the unstable topologies are prevalent.
§1. We shall denote by u α ( P ) = u α ( x 1 , x 2 , x 3 , t ), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 , x 2 , x 3 . In considering the turbulence it is natural to assume the components of the velocity u α ( P ) at every point P = ( x 1 , x 2 , x 3 , t ) of the considered domain G of the four-dimensional space ( x 1 , x 2 , x 3 , t ) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ ² α and (d u α /d x β ) ² ― are finite and bounded in every bounded subdomain of the domain G .
Article
Preface; Nomenclature; Part I. Fundamentals: 1. Introduction; 2. The equations of fluid motion; 3. Statistical description of turbulence; 4. Mean flow equations; 5. Free shear flows; 6. The scales of turbulent motion; 7. Wall flows; Part II. Modelling and Simulation: 8. Modelling and simulation; 9. Direct numerical simulation; 10. Turbulent viscosity models; 11. Reynolds-stress and related models; 12. PDF models; 13. Large-eddy simulation; Part III. Appendices; Bibliography.
Article
An appraisal is made of several subgrid scale (SGS) viscous/scalar dissipation closures via a priori analysis of direct numerical simulation data in a temporally evolving compressible mixing layer. The effects of the filter width, the compressibility level and the Schmidt number are studied for several models. Based on the scaling of SGS kinetic energy, a new formulation for SGS viscous dissipation is proposed. This yields the best overall prediction of the SGS viscous dissipation within the inertial subrange. An SGS scalar dissipation model based on the proportionality of the turbulent time scale with the scalar mixing time scale also performs the best for the filter widths in the inertial subrange. Two dynamic methods are implemented for the determination of the model coefficients. The one based on the global equilibrium of dissipation and production is shown to be more satisfactory than the conventional dynamic model.
Article
An eighth‐order filter method for a wide range of compressible flow speeds (H. C. Yee and B. Sjogreen, Proceedings of ICOSAHOM09, June 22–26, 2009, Trondheim, Norway) is employed for large eddy simulations (LES) of temporally evolving mixing layers (TML) for different convective Mach numbers (M c ) and Reynolds numbers. The high‐order filter method is designed for accurate and efficient simulations of shock‐free compressible turbulence, turbulence with shocklets, and turbulence with strong shocks with minimum tuning of scheme parameters. The value of the M c considered is for the TML range from the quasi‐incompressible regime to the highly compressible supersonic regime. The three main characteristics of compressible TML (the self‐similarity property, compressibility effects, and the presence of large‐scale structures with shocklets for high M c ) are considered for the LES study. The LES results that used the same scheme parameters for all studied cases agree well with experimental results and published direct numerical simulations (DNS). Published 2012. This article is a US Government work and is in the public domain in the USA.
Article
The occurrence of shocks in the confined three‐dimensional turbulent mixing layer at convective Mach number 1.2 is established by means of direct numerical simulations. The shocks are generated by the turbulent motions in the flow. Consequently, they can have different shapes and orientations, while they persist for a relatively short time. Furthermore, they are created by different types of turbulent vortices. The shocks do not strongly contribute to the turbulent dissipation. Even at the time when the largest shocks occur, the fraction of the turbulent dissipation due to the shocks is less than 10%.
Article
The turbulent flow originating from the interaction between two parallel streams with different velocities is studied by means of direct numerical simulation. Rather than the more common temporal evolving layer, a spatially evolving configuration, with perturbed laminar inlet conditions is considered. The streamwise evolution and the self-similar state of turbulence statistics are reported and compared to results available in the literature. The characteristics of the transitional region agree with those observed in other simulations and experiments of mixing layers originating from laminar inlets. The present results indicate that the transitional region depends strongly on the inlet flow. Conversely, the self-similar state of turbulent kinetic energy and dissipation agrees quantitatively with those in a temporal mixing layer developing from turbulent initial conditions [M. M. Rogers and R. D. Moser, “Direct simulation of a self-similar turbulent mixing layer,” Phys. Fluids 6, 903 (1994)]. The statistical features of turbulence in the self-similar region have been analysed in terms of longitudinal velocity structure functions, and scaling exponents are estimated by applying the extended self-similarity concept. In the small scale range (60 < r/η < 250), the scaling exponents display the universal anomalous scaling observed in homogeneous isotropic turbulence. The hypothesis of isotropy recovery holds in the turbulent mixing layer despite the presence of strong shear and large-scale structures, independently of the means of turbulence generation. At larger scales (r/η > 400), the mean shear and large coherent structures result in a significant deviation from predictions based on homogeneous isotropic turbulence theory. In this second scaling range, the numerical values of the exponents agree quantitatively with those reported for a variety of other flows characterized by strong shear, such as boundary layers, as well as channel and wake flows.