Effects of nozzle-exit boundary-layer profile on the initial shear-layer instability, flow field and noise of subsonic jets

The influence of the nozzle-exit boundary-layer profile on high-subsonic jets is investigated by performing compressible large-eddy simulations (LES) for three isothermal jets at a Mach number of 0.9 and a diameter-based Reynolds number of $5\times 10^{4}$, and by conducting linear stability analyses from the mean-flow fields. At the exit section of a pipe nozzle, the jets exhibit boundary layers of momentum thickness of approximately 2.8 % of the nozzle radius and a peak value of turbulence intensity of 6 %. The boundary-layer shape factors, however, vary and are equal to 2.29, 1.96 and 1.71. The LES flow and sound fields differ significantly between the first jet with a laminar mean exit velocity profile and the two others with transitional profiles. They are close to each other in these two cases, suggesting that similar results would also be obtained for a jet with a turbulent profile. For the two jets with non-laminar profiles, the instability waves in the near-nozzle region emerge at higher frequencies, the mixing layers spread more slowly and contain weaker low-frequency velocity fluctuations and the noise levels in the acoustic field are lower by 2–3 dB compared to the laminar case. These trends can be explained by the linear stability analyses. For the laminar boundary-layer profile, the initial shear-layer instability waves are most strongly amplified at a momentum-thickness-based Strouhal number $St_{\unicode[STIX]{x1D703}}=0.018$, which is very similar to the value obtained downstream in the mixing-layer velocity profiles. For the transitional profiles, on the contrary, they predominantly grow at higher Strouhal numbers, around $St_{\unicode[STIX]{x1D703}}=0.026$ and 0.032, respectively. As a consequence, the instability waves rapidly vanish during the boundary-layer/shear-layer transition in the latter cases, but continue to grow over a large distance from the nozzle in the former case, leading to persistent large-scale coherent structures in the mixing layers for the jet with a laminar exit velocity profile.

Numerical study of turbulent separation bubbles with varying pressure gradient and Reynolds number

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

A New Method for Scaling Submarine Resistance Model Data

A new method is presented for obtaining full-scale resistance predictions from model-scale submarine resistance tests. Traditionally, model-scale submarine resistance data is scaled using Froude’s Hypothesis, where the resistance is decomposed into frictional resistance and residual resistance. The frictional resistance is computed from a flat plate friction line formula at both model- and full-scale. The residual resistance coefficient is measured model-scale and is assumed to be independent of Reynolds number. Since the flow is assumed to be turbulent at fullscale, turbulence stimulators are placed at about 5% x/L to trip the flow in the model tests. The parasitic drag from the turbulence stimulators must be measured and subtracted from the model-scale resistance. In the new approach a momentum thickness simulator is used in place of a traditional turbulence stimulator. The purpose of the momentum thickness simulator is to achieve similarity for model-scale and full-scale normalized momentum thickness of the boundary layer at the stern of the vessel as momentum thickness is directly related to ship resistance. The size and location of the momentum thickness simulator is computed to assure the flow is turbulent downstream of the simulator and to produce similar momentum thickness on the body downstream of the simulator. With this approach the total resistance is scaled from model-scale to full-scale without decomposing the resistance into frictional resistance and residual resistance. The parasitic drag is not measured and subtracted, as the additional drag from the momentum simulator is computed to produce the correctly scaled momentum thickness. In the new approach the physics associated with resistance is more clearly expressed and improved accuracy of resistance predictions is expected.

A Relatively Simple Integral Method for Turbulent Flow Over Rough Surfaces

The integral form of the equation for x momentum is solved for the skin friction coefficient, in external thin boundary layer flow, on surfaces whose technical roughness elements' size is given. This is done by using a “roughness depression function” in the law of the wall and wake which serves as the needed velocity profile. The method uses the equivalent sand grain size concept in its calculations. Predictions are made of the friction coefficient, Cf, as a function of momentum thickness Reynolds number and also, of Cf's dependence on the ratio of momentum thickness to the size of the technical (actual) roughness elements. In addition, boundary layer thicknesses and velocity profiles on rough surfaces are calculated and, when available, comparisons are made with the experimental data from a number of sources in the literature. Also, comparisons are made with the results of another major predictive scheme which does not use the equivalent sand grain concept.

Data-enabled prediction of streak breakdown in pressure-gradient boundary layers

Streaks in pre-transitional boundary layers are analysed and their properties are extracted from direct numerical simulation data. Streaks that induce breakdown to turbulence via secondary instability are shown to differ from the remainder of the population in various attributes. Conditionally averaged flow fields establish that they are situated farther away from the wall, and have a larger cross-sectional area and higher peak amplitude. The analysis also shows that the momentum thickness acts as a similarity parameter for the properties of the streaks. Probability density functions of the streak amplitude, area, and shear along the streaks, collapse among the various pressure gradients when plotted as a function of the momentum thickness. A prediction scheme for laminar–turbulent transition based on artificial neural networks is presented, which can identify the streaks that will eventually induce the formation of turbulent spots. In comparison to linear stability theory, the approach achieves a higher prediction accuracy at considerably lower computational cost.

Direct numerical simulation of the incompressible temporally developing turbulent boundary layer

We perform a direct numerical simulation (DNS) investigation of the incompressible temporally developing turbulent boundary layer. The approach is inspired by temporal simulations of flows which are generally thought of as developing in space, such as wakes and mixing layers. Compressible boundary layers have previously been studied in this manner yet the temporal approach appears to be under-exploited in the literature concerning incompressible boundary layers. The flow is the turbulent counterpart to the laminar Rayleigh problem or Stokes’ first problem, in which a fluid at rest is set into motion by a wall moving at constant velocity. An initial profile that models the effect of a wall-mounted trip wire is implemented and allows the characterisation of initial conditions by a trip Reynolds number. For the current set-up, a trip Reynolds number of 500 based on the trip-wire diameter successfully triggers transition yet only mildly perturbs the flow so it assumes a natural development at the lowest possible Reynolds number based on momentum thickness. A systematic trip study reveals that as the ratio of momentum thickness to trip-wire diameter approaches unity, our flow approaches a state free from the effects of its starting trip Reynolds number. The transport of a passive scalar by this flow is also simulated. The role played by domain size is investigated with two boxes, sized to accommodate two chosen final Reynolds numbers. Comparisons of the skin friction coefficient, velocity and scalar statistics demonstrate that the temporally developing boundary layer is a good model for the spatially developing boundary layer once initial conditions can be neglected. Analysis of similarity solutions suggests such a rapprochement of the spatial and temporal boundary layers may be expected at high Reynolds numbers given that the only terms that asymptotically persist are those common to both cases. If one seeks statistics for the turbulent boundary layer, the temporal boundary layer is therefore a viable method if modest convergence is sufficient. We suggest that such a temporal set-up could prove useful in the study of turbulence dynamics.

Very near-nozzle shear-layer turbulence and jet noise

One of the principal challenges in the prediction and design of low-noise nozzles is accounting for the near-nozzle turbulent mixing layers at the high Reynolds numbers of engineering conditions. Even large-eddy simulation is a challenge because the locally largest scales are so small relative to the nozzle diameter. Model-scale experiments likewise typically have relatively thick near-nozzle shear layers, which potentially hampers their applicability to high-Reynolds-number design. To quantify the sensitivity of the far-field sound to nozzle turbulent-shear-layer conditions, a family of diameter $D$ nozzles is studied in which the exit turbulent boundary layer momentum thickness is varied from $0.0042D$ up to $0.021D$ for otherwise identical flow conditions. Measurements include particle image velocimetry (PIV) to within $0.04D$ of the exit plane and far-field acoustic spectra. The influence of the initial turbulent-shear-layer thickness is pronounced, though it is less significant than the well-known sensitivity of the far-field sound to laminar versus turbulent shear-layer exit conditions. For thicker shear layers, the nominally missing region, where the corresponding thinner shear layer would develop, leads to the noise difference. The nozzle-exit momentum thickness successfully scales the high-frequency radiated sound for nozzles of different sizes and exhaust conditions. Yet, despite this success, the detailed turbulence statistics show distinct signatures of the different nozzle boundary layers from the different nozzles. Still, the different nozzle shear-layer thicknesses and shapes have a similar downstream development, which is consistent with a linear stability analysis of the measured velocity profiles.

An Integral Solution for Skin Friction in Turbulent Flow Over Aerodynamically Rough Surfaces With an Arbitrary Pressure Gradient

The combined law of the wall and wake, with the inclusion of the “roughness depression function” for the inner law in the “Log” region, is used as the inner coordinates' velocity profile in the integral form of the x momentum equation to solve for the local skin friction coefficient. The “equivalent sand grain roughness” concept is employed in the roughness depression function in the solution. Calculations are started at the beginning of roughness on a surface, as opposed to starting them using the measured experimental values at the first data point, when making comparisons of predictions with data sets. The dependence of the velocity wake strength on both pressure gradient and momentum thickness Reynolds number are taken into account. Comparisons of the prediction with experimental skin friction data, from the literature, have been made for some adverse, zero, and favorable (accelerating flows) pressure gradients. Predictions of the shape factor, roughness Reynolds number, and momentum thickness Reynolds number and comparisons with data are also made for some cases. In addition, some comparisons with the predictions of earlier investigators have also been made.

Large Eddy Simulations of a Stratified Shear Layer

The performance of the large eddy simulation (LES) approach in predicting the evolution of a shear layer in the presence of stratification is evaluated. The LES uses a dynamic procedure to compute subgrid model coefficients based on filtered velocity and density fields. Two simulations at different Reynolds numbers are simulated on the same computational grid. The fine LES simulated at a low Reynolds number produces excellent agreement with direct numerical simulations (DNS): the linear evolution of momentum thickness and bulk Richardson number followed by an asymptotic approach to constant values is correctly represented and the evolution of the integrated turbulent kinetic energy budget is well captured. The model coefficients computed from the velocity and the density fields are similar and have a value in range of 0.01-0.02. The coarse LES simulated at a higher Reynolds number Re = 50,000 shows acceptable results in terms of the bulk characteristics of the shear layer, such as momentum thickness and bulk Richardson number. Analysis of the turbulent budgets shows that, while the subgrid stress is able to remove sufficient energy from the resolved velocity fields, the subgrid scalar flux and thereby the subgrid scalar dissipation are underestimated by the model.

Crossover between two- and three-dimensional turbulence in spatial mixing layers

We investigate how the domain depth affects the turbulent behaviour in spatially developing mixing layers by means of large-eddy simulations based on a spectral vanishing viscosity technique. Analyses of spectra of the vertical velocity, of Lumley’s diagrams, of the turbulent kinetic energy and of the vortex stretching show that a two-dimensional behaviour of the turbulence is promoted in spatial mixing layers by constricting the fluid motion in one direction. This finding is in agreement with previous works on turbulent systems constrained by a geometric anisotropy, pioneered by Smith, Chasnov & Waleffe (Phys. Rev. Lett., vol. 77, 1996, pp. 2467–2470). We observe that the growth of the momentum thickness along the streamwise direction is damped in a confined domain. An almost fully two-dimensional turbulent behaviour is observed when the momentum thickness is of the same order of magnitude as the confining scale.