A numerical study of a class of TVD schemes for compressible mixing layers

At high Mach numbers the two-dimensional time-developing mixing layer develops shock waves, positioned around large-scale vortical structures. A suitable numerical method has to be able to capture the inherent instability of the flow, leading to the roll-up of vortices, and also must be able to capture shock waves when they develop. Standard schemes for low speed turbulent flows, for example spectral methods, rely on resolution of all flow-features and cannot handle shock waves, which become too thin at any realistic Reynolds number. The performance of a class of second-order explicit total variation diminishing (TVD) schemes on a compressible mixing layer problem was studied. The basic idea is to capture the physics of the flow correctly, by resolving down to the smallest turbulent length scales, without resorting to turbulence or sub-grid scale modeling, and at the same time capture shock waves without spurious oscillations. The present study indicates that TVD schemes can capture the shocks accurately when they form, but (without resorting to a finer grid) have poor accuracy in computing the vortex growth. The solution accuracy depends on the choice of limiter. However a larger number of grid points are in general required to resolve the correct vortex growth. The low accuracy in computing time-dependent problems containing shock waves as well as vortical structures is partly due to the inherent shock-capturing property of all TVD schemes. In order to capture shock waves without spurious oscillations these schemes reduce to first-order near extrema and indirectly produce clipping phenomena, leading to inaccuracy in the computation of vortex growth. Accurate simulation of unsteady turbulent fluid flows with shock waves will require further development of efficient, uniformly higher than second-order accurate, shock-capturing methods

The effect of Mach number on unstable disturbances in shock/boundary-layer interactions

The effect of Mach number on the growth of unstable disturbances in a boundary layer undergoing a strong interaction with an impinging oblique shock wave is studied by direct numerical simulation and linear stability theory (LST). To reduce the number of independent parameters, test cases are arranged so that both the interaction location Reynolds number (based on the distance from the plate leading edge to the shock impingement location for a corresponding inviscid flow) and the separation bubble length Reynolds number are held fixed. Small-amplitude disturbances are introduced via both white-noise and harmonic forcing and, after verification that the disturbances are convective in nature, linear growth rates are extracted from the simulations for comparison with parallel flow LST and solutions of the parabolized stability equations (PSE). At Mach 2.0, the oblique modes are dominant and consistent results are obtained from simulation and theory. At Mach 4.5 and Mach 6.85, the linear Navier-Stokes results show large reductions in disturbance energy at the point where the shock impinges on the top of the separated shear layer. The most unstable second mode has only weak growth over the bubble region, which instead shows significant growth of streamwise structures. The two higher Mach number cases are not well predicted by parallel flow LST, which gives frequencies and spanwise wave numbers that are significantly different from the simulations. The PSE approach leads to good qualitative predictions of the dominant frequency and wavenumber at Mach 2.0 and 4.5, but suffers from reduced accuracy in the region immediately after the shock impingement. Three-dimensional Navier-Stokes simulations are used to demonstrate that at finite amplitudes the flow structures undergo a nonlinear breakdown to turbulence. This breakdown is enhanced when the oblique-mode disturbances are supplemented with unstable Mack modes

Use of passive scalar tagging for the study of coherent structures in the plane mixing layer

Data obtained from the numerical simulation of a 2-D mixing layer were used to study the feasibility of using the instantaneous concentration of a passive scalar for detecting the typical coherent structures in the flow. The study showed that this technique works quite satisfactorily and yields results similar to those that can be obtained by using the instantaneous vorticity for structure detection. Using the coherent events educed by the scalar conditioning technique, the contribution of the coherent events to the total turbulent momentum and scalar transport was estimated. It is found that the contribution from the typical coherent events is of the same order as that of the time-mean value. However, the individual contributions become very large during the pairing of these structures. The increase is particularly spectacular in the case of the Reynolds shear stress

Performance of Low Dissipative High Order Shock-Capturing Schemes for Shock-Turbulence Interactions

Accurate and efficient direct numerical simulation of turbulence in the presence of shock waves represents a significant challenge for numerical methods. The objective of this paper is to evaluate the performance of high order compact and non-compact central spatial differencing employing total variation diminishing (TVD) shock-capturing dissipations as characteristic based filters for two model problems combining shock wave and shear layer phenomena. A vortex pairing model evaluates the ability of the schemes to cope with shear layer instability and eddy shock waves, while a shock wave impingement on a spatially-evolving mixing layer model studies the accuracy of computation of vortices passing through a sequence of shock and expansion waves. A drastic increase in accuracy is observed if a suitable artificial compression formulation is applied to the TVD dissipations. With this modification to the filter step the fourth-order non-compact scheme shows improved results in comparison to second-order methods, while retaining the good shock resolution of the basic TVD scheme. For this characteristic based filter approach, however, the benefits of compact schemes or schemes with higher than fourth order are not sufficient to justify the higher complexity near the boundary and/or the additional computational cost

Direct numerical simulation of compressible turbulence in a counter-flow channel configuration

Counter-flow configurations, whereby two streams of fluid are brought together from opposite directions, are highly efficient mixers due to the high turbulence intensities that can be maintained. In this paper, a simplified version of the problem is introduced that is amenable to direct numerical simulation. The resulting turbulent flow problem is confined between two walls, with one non-zero mean velocity component varying in the space direction normal to the wall, corresponding to a simple shear flow. Compared to conventional channel flows, the mean flow is inflectional and the maximum turbulence intensity relative to the maximum mean velocity is nearly an order of magnitude higher. The numerical requirements and turbulence properties of this configuration are first determined. The Reynolds shear stress is required to vary linearly by the imposed forcing, with a peak at the channel centreline. A similar behaviour is observed for the streamwise Reynolds stress, the budget of which shows an approximately uniform distribution of dissipation, with large contributions from production, pressure-strain and turbulent diffusion. A viscous sublayer is obtained near the walls and with increasing Reynolds number small-scale streaks in the streamwise momentum are observed, superimposed on the large-scale structures that buffet this region. When the peak local mean Mach number reaches 0.55, turbulent Mach numbers of 0.6 are obtained, indicating that this flow configuration can be useful to study compressibility effects on turbulence

Two-dimensional unsteadiness map of oblique shock wave/boundary layer interaction with sidewalls

The low-frequency unsteadiness of oblique shock wave/boundary layer interactions (SBLIs) has been investigated using large-eddy simulation (LES) and high-frequency pressure measurements from experiments. Particular attention has been paid to off-centreline behaviour: the LES dataset was generated including sidewalls and experimental pressure measurements were acquired across the entire span of the reflected shock foot. The datasets constitute the first maps of low-frequency unsteadiness in both streamwise and spanwise directions. The results reveal that significant low-frequency shock motion (with St ≈ 0.03) occurs away from the centreline, along most of the central separation shock and in the corner regions. The most powerful low frequency unsteadiness occurs offcentre, likely due to the separation shock being strengthened by shocks arising from the swept interactions on the sidewalls. Both simulation and experimental results exhibit asymmetry about the spanwise centre. In simulations, this may be attributed to a lack of statistical convergence; however, the fact that this is also seen in experiments is indicative that some SBLIs may exhibit some inherent asymmetry across the two spanwise halves of the separation bubble. There is also significant low-frequency power in the corner separations. The relation of the unsteadiness in the corner regions to that in the centre is investigated by means of two-point correlations: a key observation is that significant correlation does not extend across the attached flow channel between the central and corner separations

Strong interaction of a turbulent spot with a shock-induced separation bubble

Direct numerical simulations have been conducted to study the passage of a turbulent spot through a shock-induced separation bubble. Localized blowing is used to trip the boundary layer well upstream of the shock impingement, leading to mature turbulent spots at impingement, with a length comparable to the length of the separation zone. Interactions are simulated at free stream Mach numbers of two and four, for isothermal (hot) wall boundary conditions. The core of the spot is seen to tunnel through the separation bubble, leading to a transient reattachment of the flow. Recovery times are long due to the influence of the calmed region behind the spot. The propagation speed of the trailing interface of the spot decreases during the interaction and a substantial increase in the lateral spreading of the spot was observed. A conceptual model based on the growth of the lateral shear layer near the wingtips of the spot is used to explain the change in lateral growth rat

Applying a transdisciplinary mixed methods research design to explore sustainable diets in rural South Africa

This article describes a conceptual framework for exploring sustainable diets, using a case study example of ongoing research in the Vaalharts region, a rural setting in South Africa. A qualitative research approach is followed with an integrated transdisciplinary mixed methods research design with multiple concurrent components employed during two sequential phases. A successful application of the framework is achieved through a collaborative team effort of researchers with qualitative and quantitative research expertise transcending different disciplines, as well as participation of community members throughout the research process. We demonstrate that transdisciplinary mixed methods research designs are essential to gain a better understanding of the complex concept of sustainable diets

Change in drag, apparent slip and optimum air layer thickness for laminar flow over an idealised superhydrophobic surface

Analytic results are derived for the apparent slip length, the change in drag and the optimum air layer thickness of laminar channel and pipe flow over an idealised superhydrophobic surface, i.e. a gas layer of constant thickness retained on a wall. For a simple Couette flow the gas layer always has a drag reducing effect, and the apparent slip length is positive, assuming that there is a favourable viscosity contrast between liquid and gas. In pressure-driven pipe and channel flow blockage limits the drag reduction caused by the lubricating effects of the gas layer; thus an optimum gas layer thickness can be derived. The values for the change in drag and the apparent slip length are strongly affected by the assumptions made for the flow in the gas phase. The standard assumptions of a constant shear rate in the gas layer or an equal pressure gradient in the gas layer and liquid layer give considerably higher values for the drag reduction and the apparent slip length than an alternative assumption of a vanishing mass flow rate in the gas layer. Similarly, a minimum viscosity contrast of four must be exceeded to achieve drag reduction under the zero mass flow rate assumption whereas the drag can be reduced for a viscosity contrast greater than unity under the conventional assumptions. Thus, traditional formulae from lubrication theory lead to an overestimation of the optimum slip length and drag reduction when applied to superhydrophobic surfaces, where the gas is trapped