Plane Shear Layer Research Articles

Growth-Rate Prediction for Low-Speed Variable-Density Spatially Developing Turbulent Planar Shear Layers: Status and Prospects

Abstract Progress in analysis of the growth rate of low-speed variable-density spatially developing turbulent planar shear layers is examined from the viewpoint of global trends across the relevant parameter space. Several approaches have been shown to agree with available measurements. Here, it is noted that comparable agreement is also obtained using a simple extension of the canonical Kelvin–Helmholtz linear stability theory. Beyond the range of the experimentally explored parameter space, available predictions disagree qualitatively as well as quantitatively. To a degree that is not generally recognized, this indicates that the governing parameter dependences and associated phenomenology are open questions. Further investigation needed to address these questions is discussed.

Effect of tabs on the shear layer dynamics of a jet in cross-flow

Direct numerical simulations (DNS) of a jet in cross-flow (JICF) with a triangular tab at two positions are performed at jet-to-cross-flow velocity ratios of $R = 2$ and $4$ with a jet Reynolds number of 2000 based on the jet's bulk velocity and exit diameter. The DNS and dynamic mode decomposition show the sensitivity of the tab's effect on the jet upstream shear layer (USL) structure and cross-section to $R$ , echoing the experimental discoveries of Harris et al. (J. Fluid Mech., vol. 918, 2021). Furthermore, DNS reveals that the presence of a tab placed on the upstream side of the nozzle significantly modifies the USL through production of streamwise vortices that curl around the spanwise vortex tubes originating from the primary instability of the USL. This provides an explanation for the improvement in mixing that has been associated with an upstream tab. The streamwise vortex structure shows remarkable similarities to the ‘strain-oriented vortex tubes’ observed for disturbed plane shear layers by Lasheras & Choi (J. Fluid Mech., vol. 189, 1988, pp. 53–86). For both $R$ cases, the USL instability is delayed, the jet penetration is reduced, and the jet cross-section is flattened, although the tab has a less pronounced effect on the USL structure at higher velocity ratios, where the formation of the streamwise vortices is delayed. In contrast, a tab placed 45 $^\circ$ from the upstream position produces significantly different effects compared with the upstream tab. At $R = 4$ , the jet cross-section is significantly skewed away from the tab and a tertiary vortex is formed, as observed in past studies of round JICFs at relatively high $R$ and low Reynolds numbers. The ability of the tab to produce a controllable steady-state tertiary vortex has implications for a variety of applications. The 45 $^\circ$ tab produces asymmetric effects in the wake of the jet at $R = 2$ , but the effect on the jet cross-section is much smaller, highlighting the sensitivity of jets at high $R$ to asymmetric perturbations.

A unified theory for the envelope radiation of ring-mode coherent structures in the very-near-nozzle and developed regions of a circular jet

Large-scale coherent structures (CS) are present on a turbulent circular jet and are known to constitute an important source of jet noise. These structures may be treated as wavepackets of instability modes supported by the time-averaged mean flow. The nonlinear evolution and acoustic radiation of such CS in the fully developed region of a jet have been described in our recent work [Zhang and Wu, “Nonlinear evolution and low-frequency acoustic radiation of ring-mode coherent structures on subsonic turbulent circular jets,” J. Fluid Mech. 940(A39), 1–56 (2022)], where the mechanism of envelope radiation, or low-frequency radiation, was identified. The present paper focuses on the distinguished very-near-nozzle region, where the nozzle diameter is comparable with the length scale of the envelope of the CS. The CS resides in the thin shear layer developing from the lip line, and its nonlinear dynamics resembles that on a planar shear layer, only slightly affected by circular geometry, or circularity. However, circularity causes a leading-order effect on the acoustic radiation. One of significant new features is the presence of an inner acoustic field in the potential core. The outer and inner acoustic fields interact with each other, and the equivalent sources of sound have to be determined along with the sound fields. The waves in the inner acoustic field propagate mainly upstream. It is shown for the first time that nonlinear interactions of CS in the thin shear layer excite via receptivity the so-called discrete “trapped acoustic modes” as well as low-frequency Kelvin–Helmholtz modes, which may evolve into preferred modes in the developed region. The envelope radiation provides a key mechanism coupling the dynamics in the thin shear layer with the dynamics in the jet column regions. As the jet develops downstream, the inner acoustic field narrows gradually and disappears eventually. It is demonstrated that the theory developed here for nonlinear dynamics and acoustic radiation is valid from the very-near-nozzle to the developed regions, and thus stands as a unified theory.

Three-dimensional, rotational flamelet closure model with two-way coupling

A new flamelet model is developed to be used for subgrid modelling and coupled with the resolved flow description for turbulent combustion. The model differs from current models in several critical ways. (i) Non-premixed flames, premixed flames or multi-branched flame structures are determined rather than prescribed. (ii) The effects of shear strain and vorticity on the flames are determined. (iii) The strain rates and vorticity applied at the subgrid level are directly determined from the resolved-scale strain rates and vorticity without the use of a contrived progress variable. (iv) The flamelet model is three-dimensional without need for assuming axisymmetry or planar geometry. (v) The effect of variable density is addressed in the flamelet model. Solutions to the Navier–Stokes equations and the associated scalar equations governing the flamelet model are obtained without the boundary-layer approximation. By appropriate coordinate transformation, a similar solution is found for the rotational flamelet model, reducing it to a system of ordinary differential equations. Vorticity is shown to create a centrifugal force on the subgrid counterflow that modifies the molecular transport rates and burning rate. Sample computations of the rotational flamelet model without coupling to the resolved flow are presented first to demonstrate the importance of the new features of the model. Scaling laws are presented for relating strain rates and vorticity at the subgrid level to those quantities at the resolved-flow level for coupling with large-eddy simulations or the time-averaged mean-flow level for Reynolds-averaged flows. The time-averaged behaviour of a simple turbulent flow is resolved with coupling to the rotational flamelet model. Specifically, a two-dimensional, multicomponent, time-averaged planar shear layer with variable density and energy release is employed using a mixing-length description for the eddy viscosity. Needs for future study are identified.

Optimal growth over a time-evolving variable-density jet at Atwood number

Secondary instabilities growing over a time-evolving variable-density round jet subject to the primary Kelvin–Helmholtz (KH) instability at Atwood number $\vert \textit {At} \vert = 0.25$ are investigated with a non-modal linear stability analysis. Despite local modifications of the base flow vorticity induced by the baroclinic torque, these disturbances experience a short-term universal growth due to a combination of the Orr and lift-up mechanisms, whatever the azimuthal wavenumber $m$ . At $\textit {Re}=1000$ , the secondary energy growth stems from the development of elliptical and hyperbolic instabilities, with an E-type-to-H-type transition as $m$ and $\textit {Re}$ increase, as in the homogeneous case (Nastro et al., J. Fluid Mech., vol. 900, 2020, A13). In the light jet at $\textit {Re} = 1000$ , after the KH mode saturation, the high- $m$ H-type instability is replaced by a perturbation organised as counter-rotating streamwise vortices located in the base flow region of promoted strain rate. Increasing the Reynolds number up to $\textit {Re} = 10\,000$ yields larger energy growths and a strong anisotropy among energy and enstrophy components with a preferential increase of axial velocity and azimuthal vorticity. Both come from the linearised baroclinic source that drives the optimal response towards folded sheets of axial velocity that differ from those observed in the variable-density plane shear layers. When the perturbation is injected around the KH saturation time for $\textit {Re}=10\,000$ , the response to optimal perturbation takes the form of fast growing secondary KH instabilities whatever $m$ . We find these three-dimensional secondary KH instabilities to be good candidates for the transition to turbulence in variable-density jet flows.

Wake behind circular cylinder excited by spanwise non-uniform disturbances

We studied a modification of wake behind a circular cylinder using a plasma actuator. The plasma actuators were arranged in the spanwise direction of the cylinder to give temporal periodic disturbances having Strouhal number St = 0.18-2.3 with a burst ratio BR = 20 and 40%. The Reynolds number was set in a rage of Re = 4200 to 8400. Two types of plasma actuator were prepared; one is a single strip of the actuator placed at each side of the cylinder to give a spanwise uniform disturbance, and another is an array of small piece of actuators placed at the same location to create a spanwise non-uniform disturbance with temporal phase difference, φ = 0 or π, between adjacent electrodes. A conventional two-component PIV and stereo PIV was used to measure the flow field. Figure 1 shows the instantaneous spanwise component of vorticity at Re = 4200 evaluated by two-component PIV. Under no disturbance condition, the laminar shear layer extends straight to around x / d = 1.5 and then forms a wake vortex, as shown in Fig.1(a). In the case of spanwise non-uniform forcing with St = 1.09 and φ =π, rapid roll up of the initial shear layer leads to arrangement of wake vortices closer to the cylinder., as shown in Fig.1(b). With higher Strouhal number case with St = 1.09 and φ = 0, shown in Fig.1(c), a series of fine scale vortices are generated behind both side of the cylinder without forming regular Karman vortices. The spanwise non-uniform forcing was effective to suppress the formation of large scale vortices just behind the cylinder. Figure 2 shows surface of constant vorticity magnitude and vortex lines under St =1.09 and φ = π case. These were computed from a phase-averaged threecomponents velocity field evaluated by stereo PIV. The value of the surface was selected to display the boundary layer formed on the cylinder, and the vortex lines are selected to visualize the vortex structure formed in the following shear layer. A bundle of vortex lines are shaped in a wavy pattern along spanwise direction with 180 degrees out of phase to the adjacent bundle upstream of downstream. This structure, so called ‘chain-line fence structure’ was already found in planar free shear layer [Nygaard, K.J. and Glezer, A., 1990, Phys. Fluids A, 2, 461] and planar jet [Sakakibara, J., Anzai, T., 2001, Phys. Fluids, 13, 1541], but it became evident to create it in the wake of circular cylinder in this study.

The near-field aerodynamic characteristics of hot high-speed jets

Abstract

Numerical study of passive forcing on the secondary instability of a laminar planar free shear layer

Direct numerical simulations were performed of a free shear layer developing downstream from a no-slip splitter plate. The simulation results show that three-dimensional streaky perturbations developing in the boundary layer on the high-speed side of the splitter plate are convected downstream from the trailing edge and influence the development of three-dimensional perturbations in the planar free shear layer. Simulations are performed in which a spanwise-wavy shape is introduced to the trailing edge of the splitter plate to influence the downstream development of the free shear layer. Introduction of the spanwise-waviness to the trailing edge results in significantly earlier small-scale breakdown of the free shear layer.

A Unified Correction Method for the Acoustic Refraction (UCMAR) Caused by a Three Dimensional Shear Layer

The acoustic refraction induced by the shear layer in an open-jet wind tunnel causes a source shift when estimating the source location with beamforming. Traditional correction methods of the shear layer refraction are achieved through a computational eff ort or limited using one-dimensional or planar shear layer. In this paper, the unified correction method for acoustic refraction (UCMAR) is suitable for the three dimensions that covers several traditional forms. Meanwhile, the UCMAR can consider more general configurations, such as the temperature gradient on both sides of the shear layer and the off -axis source in a circular wind tunnel. These configurations are validated through a ray tracing technique and a benchmark example. In addition, the principle of time reverse is integrated with UCMAR. This results in a reverse UCMAR, which can quickly attain an acceptable solution.

Comparison of Intrusive and Non-Intrusive Mixture Composition Measurements in Supersonic Flow

Accurately quantifying mixture composition in supersonic flows is vital to the experimental development of fuel–air mixing enhancement strategies for scramjet propulsion applications. Mixture composition can be experimentally quantified by using either intrusive probe-based techniques or nonintrusive laser-based optical diagnostic techniques, with the different methods each offering unique advantages and disadvantages. To this end, the work presented herein provides a comparison of mixture composition measurements made with an intrusive gas-sampling probe to measurements of the same quantity made with the nonintrusive filtered Rayleigh scattering technique. The measurements are made downstream of a strut injector, which sonically injects helium parallel to a Mach 2.5 airflow, creating a two-dimensional planar shear layer composed of a binary mixture of helium and air. The experimental results show that the helium mole fraction measurements obtained using these two different techniques compare quite favorably. Specifically, at the primary downstream station of interest, all of the points surveyed with the gas-sampling probe lie within the 95% uncertainty bands of the filtered Rayleigh scattering measurements, with an average absolute difference in helium mole fraction of between the two techniques.

Numerical analysis on interactions of vortex, shock wave, and exothermal reaction in a supersonic planar shear layer laden with droplets

The analysis on the interactions of a large-scale shearing vortex, an incident oblique shock wave, and a chemical reaction in a planar shear layer is performed by numerical simulations. The reacting flows are obtained by directly solving the multi-species Navier-Stokes equations in the Eulerian frame, and the motions of individual point-mass fuel droplets are tracked in the Lagrangian frame considering the two-way coupling. The influences of shock strength and spray equivalence ratio on the shock-vortex interaction and the induced combustion are further studied. Under the present conditions, the incident shock is distorted by the vortex evolution to form the complicated waves including an incident shock wave, a multi-refracted wave, a reflected wave, and a transmitted wave. The local pressure and temperature are elevated by the shock impingement on the shearing vortex, which carries flammable mixtures. The chemical reaction is mostly accelerated by the refracted shock across the vortex. Two different exothermal reaction modes could be distinguished during the shock-vortex interaction as a thermal mode, due to the additional energy from the incident shock, and a local quasi detonation mode, due to the coupling of the refracted wave with reaction. The former mode detaches the flame and shock wave, whereas the latter mode tends to occur when the incident shock strength is higher and local equivalence ratio is higher approaching to the stoichiometric value. The numerical results illustrate that those two modes by shock-vortex interaction depend on the structure of the post-shock flame kernel, which may be located either in the vortex-braids of post-shock flows or in the shock-vortex interaction regime.

Investigation of large-scale structures in supersonic planar non-reactive shear layer with a base flow region

Large-scale structures in supersonic planar non-reactive shear layer with a base flow region are experimentally investigated. By utilizing the recently developed nanoparticle-based planar laser scattering (NPLS) method, the fine transient streamwise and spanwise flow fields are acquired, and the large-scale structures in the reattachment region, initial turbulent shear region, and developing turbulent shear region are analyzed. The predominant instability is convective for the two separated shear layer, whereas the instability in the base flow zone is still absolute according to the linear instability analysis. From the initial to developing turbulent shear layer region, the predominant instability transited gradually from absolute to convective instability, and the corresponding scales of the large-scale structures turn from about half to whole of the shear layer thickness. The entire streamwise mean flow topology and the streamwise and transverse velocity fluctuations are compared with different pressure ratios and with incompressible blunt trailing-edge shear layer. Linear instability analysis shows further evidence to the instability evolution of the supersonic shear layer with a base flow along the streamwise. It implied that instability induced by unmatched pressure in the wake zone can increase the absolute instability and improve the receptivity to inflow perturbations. The reattachment points are approximated with the intersection points of the reattachment shock pairs, which can be obtained from the transient NPLS images. The distribution of the reattachment points suggests that: the principal axis of the reattachment point distribution is dominated by the high-speed flow, and unmatched pressure increases the perturbation in the wake zone, result in a high absolute instability.

A fast ray casting method for sound refraction at shear layers

The application of microphone arrays for aeroacoustic measurements in wind tunnels with an open test section requires to consider the sound refraction at the shear layer. Several methods are available for this that are either applicable to special scenarios such as planar or cylindrical thin shear layers or that require considerable computational effort. The paper concerns a new method for general application—ray casting—that is based on a ray tracing method. Ray tracing is briefly revisited and it is shown that the computational effort becomes quite high especially for large mapping grids. The ray casting approach with its interpolation technique is introduced. A time reversal technique is the key to make this approach very efficient and fast. Finally, the approach is demonstrated using two example results from simulation and from a practical measurement.

Turbulent mixing and entrainment in a stratified horizontal plane shear layer: joint velocity–temperature analysis of experimental data

Buoyancy effects on the turbulent mixing and entrainment processes were analysed in the case of a stratified plane shear layer between two horizontal air flows in conditions leading to relatively low values of the flux Richardson number ($|Ri_{f}|_{max}\simeq 0.02$). Velocity and temperature measurements were made with a single$\times$-wire probe thermo-anemometry technique, using multi-overheat sequences to deliver simultaneous velocity–temperature data at high frequency. The spatial resolution was found to be fine enough, in relation to the dissipative scale and the thermal diffusive scale, to avoid false mixing enhancement in the analysis of the physical mechanisms through velocity–temperature coupling in statistical turbulence quantities. Probability density functions (PDFs) and joint probability density functions (JPDFs) were used to distinguish between the different mechanisms involved in turbulent mixing, namely entrainment, engulfing, nibbling and mixing, and point to the contribution of entrainment in the mixing process. When comparing an unstably stratified configuration to its stably stratified equivalent, no significant difference could be seen in the PDF and JPDF quantities, but a conditional analysis based on temperature thresholding enabled a separation between mixed fluid and two distinct sets of events associated with unmixed fluid entrained from the hot and cold sides of the mixing layer into the mixing layer. This separation allowed a direct calculation of the entrainment velocities on both sides of the mixing layer. A significant increase of the total entrainment could be seen in the case of unstably stratified configuration. The entrainment ratios were compared to their prediction by the Dimotakis model and both a rather good relevance of the model and some need for improvement were found from the comparison. It was hypothesised that the improvement should come from better taking into account the distinct contributions of nibbling and engulfing inside the process of entrainment and mixing.

Shear-rate-dependent viscosities of nematic and disc-like particles perfectly aligned in arbitrary relative directions to the shear layer

The shear-rate-dependent viscosity coefficients of hard ellipsoids are derived for arbitrary relative orientations to the shear flow geometry. Each of the nine components of the coefficients scaled by the Newtonian shear viscosity agrees with the previous result for the viscosity in the zero-shear-rate limit. Among the components, the shear viscosity is graphically displayed for nematic ellipsoids with an axis ratio greater than 1 and for disc-like ellipsoids with an axis ratio less than 1 over the whole range of relative orientations of the axis of ellipsoid to the plane shear layer. As the primary axis of the ellipsoid rotates from the x-axis to the y-axis in the x-y plane, the shear viscosity increases or decreases for nematic or disc-like ellipsolids, respectively.

Measurements in the annular shear layer of high subsonic and under-expanded round jets

An experimental study has been undertaken to document compressibility effects in the annular shear layers of axisymmetric jets. Comparison is made of the measured flow development with the well-documented influence of compressibility in planar mixing layers. High Reynolds number (~106) and high Mach number jets issuing from a convergent nozzle at nozzle pressure ratios (NPRs) from 1.28 to 3.0 were measured using laser Doppler anemometry instrumentation. Detailed radial profile data are reported, particularly within the potential core region, for mean velocity, turbulence rms, and turbulence shear stress. For supercritical NPRs the presence of the pressure waves in the inviscid shock cell region as the jet expanded back to ambient pressure was found to exert a noticeable effect on shear layer location, causing this to shift radially outwards at high supercritical NPR conditions. After a boundary layer to free shear layer transition zone, the turbulence development displayed a short region of similarity before adjustment to near-field merged jet behaviour. Peak turbulence rms reduction due to compressibility was similar to that observed in planar layers with radial rms suppression much stronger than axial. Comparison of the compressibility-modified annular shear layer growth rate with planar shear layer data on the basis of the convective Mach number (M C) showed notable differences; in the annular shear layer, compressibility effects began at lower M C and displayed a stronger reduction in growth. For high Mach number aerospace propulsion applications involving round jets, the current measurements represent a new data set for the calibration/validation of compressibility-affected turbulence models.

Passive scalar mixing in Mc

We investigate passive scalar mixing in spatially developing supersonic shear layer flows formed by two planar streams by means of large eddy simulation. After validation of the numerical procedures employing a high-order hybrid WENO/compact scheme by comparing the results of simulation with the results obtained from a well characterized experimental case, the effects of the convective Mach number, Mc, flow velocity ratio, r = U1/U2, and fluid density ratio, s = ρ1/ρ2, between Stream 1 and Stream 2 on passive scalar mixing are considered by examining the variation of mixing layer thickness and the mixing efficiency represented by the transport of the passive scalar. Mc is specified from 0.2 to 0.8. The evolution of large-scale coherent structures is well reproduced, with vortices undergoing rolling up, pairing, merging, and breaking up. The mixing layer thickness and mixing efficiency both decrease as Mc increases. As s increases, the mixing layer thickness increases, and the mixing efficiency of an entrained fluid decreases. As r increases, the mixing layer thickness decreases while the mixing efficiency increases. While the present results are not applicable to the examination of micro-mixing properties, which require assessment by experimental methods or high computing cost direct numerical simulation, they are useful for evaluating the effects of different flow parameters on macro-performance, which is equally important for scramjet combustor design and evaluation in engineering.

Experimental study of the mean wake of a tidal stream rotor in a shallow turbulent flow

The mean wake of a three-bladed horizontal axis tidal stream turbine operating at maximum power coefficient has been investigated experimentally in a wide flume with width 11 times the depth, providing minimal restriction to transverse wake development and behaviour of large-scale horizontal turbulence structures. This is an important first stage for understanding wake interaction in turbine arrays and hence large-scale power generation. The rotor diameter has a typical value of 60% of the depth and the thrust coefficient is representative of a full-scale turbine. The shear layers originating from the rotor tip circumference show classic linear expansion downstream, with the rate of a plane shear layer vertically and 1.5 times that horizontally. These shear layers merge by around 2.5 diameters downstream forming a self-similar two-dimensional wake beyond eight diameters downstream with a virtual origin at two diameters downstream of the rotor plane. The spreading rate is somewhat less than that for solid bodies. The detailed velocity measurements made in the near wake show rotation and vorticity similar to that measured previously for wind and marine turbines although with asymmetry associated with bed and surface proximity. The longitudinal circulation in a transverse plane is conserved at about 1% of the swept circulation from the blade tip within two diameters downstream, the extent of detailed measurement. Turbines are usually designed using blade element momentum theory in which velocities at the rotor plane are characterised by axial and tangential induction factors and it is now possible to see how this idealisation relates to actual velocities. The axial induction factor corresponds to velocity deficits at 0.4–0.8 radii from the rotor axis across the near wake while the tangential induction factor at the rotor plane corresponds to velocities at 0.4–0.6 radii between 1–2 diameters downstream, indicating some general correspondence. For the two-dimensional self-similar far wake the two parameters defining the centreline velocity deficit and the transverse velocity profiles are likely to be insensitive to Reynolds number in turbulent conditions.

On the influence of internal density variations on the linear stability characteristics of planar shear layers

Planar shear layers are known to be sensitive to density variation within the vorticity region. The current study will expand on the available literature on the stability characteristics of shear layers with density variation through consideration of new regimes in terms of the velocity and density profiles. Both spatial and spatio-temporal linear analyses were considered. In addition to the maximum density change within the shear layer, the effect of density to vorticity thickness ratio, and transverse shifting of the density profile relative to the velocity profile were considered. For a shear layer experiencing combustion, these effects relate to the relative position and thickness of the flame in the shear layer. Thick density profiles have a stabilizing affect that imposes essentially a low-pass attenuator modulation on the spatial stability. The shear layer is destabilized through the presence of an intense yet thin density reduction that overlaps with the peak vorticity profile, a condition that may be present near the shear layer origination point. The velocity ratio at which the flow transitions between absolute and convective instability is dependent on the magnitude of the density reduction in the shear layer as well as the relative density profile thickness and transverse location. The results generally indicate that the instability of the planar shear layer is strongly influenced by the characteristics of the density-weighted vorticity distribution within the shear layer.

Vortex development behind a finite porous obstruction in a channel

AbstractThis experimental study describes the turbulent wake behind a two-dimensional porous obstruction, consisting of a circular array of cylinders. The cylinders extend from the channel bed through the water surface, mimicking a patch of emergent vegetation. Three patch diameters ($D$) and seven solid volume fractions ($\Phi $) are tested. Because flow can pass through the patch, directly downstream there is a region of steady, non-zero, streamwise velocity, ${U}_{1} $, called the steady wake. For the patch diameters and solid volume fractions considered here, ${U}_{1} $ is a function of $\Phi $ only. The length of the steady wake (${L}_{1} $) increases as $\Phi $ decreases and can be predicted from the growth of a plane shear layer. The formation of the von-Kármán vortex street is delayed until the end of the steady wake. There are two regions of elevated transverse velocity fluctuation (${v}_{\mathit{rms}} $): directly behind the patch, associated with the wake turbulence of individual cylinders; and at the distance ${L}_{1} $ from the patch, associated with the formation of large-scale wake oscillation. Velocity along the centreline of the wake starts to increase only after the patch-scale vortex street is formed, and it approaches the free-stream velocity over a distance ${L}_{2} $. The dimensionless length of the entire wake, $({L}_{1} + {L}_{2} )/ D$, increases with patch porosity.