Global stability analysis of a 90°-bend pipe flow

The present work investigates the stability properties of the flow in a 90°-bend pipe with curvature $\delta =R/R_c=1/3$, with R being the radius of the cross-section of the pipe and $R_c$ the radius of curvature at the pipe centreline. Direct numerical simulations (DNS) for values of the bulk Reynolds number ${\mathit{Re}}_b=U_{b}D/\nu$ between 2000 and 3000 are performed. The bulk Reynolds number is based on the bulk velocity $U_b$, the pipe diameter D, and the kinematic viscosity $\nu$. The flow is found to be steady for ${\mathit{Re}}_b⩽2500$, with two main pairs of symmetric, counter-rotating vortices in the section of the pipe downstream of the bend. The presence of two recirculation regions is detected inside the bend: one on the outer wall and the other on the inner side. For ${\mathit{Re}}_b⩾2550$, the flow exhibits a periodic behaviour, oscillating with a fundamental non-dimensional frequency $\mathit{St}=\mathit{fD}/U_b=0.23$. A global stability analysis is performed in order to determine the cause of the transition from the steady to the periodic regime. The spectrum of the linearised Navier-Stokes operator reveals a pair of complex conjugate eigenvalues with positive real part, hence the transition is ascribed to a Hopf bifurcation occurring at ${\mathit{Re}}_{b,\mathit{cr}}\approx 2531$, a value much lower than the critical Reynolds number for the flow in a torus with the same curvature. The velocity components of the unstable direct and adjoint eigenmodes are investigated, and they display a large spatial separation, most likely due to the non-normality of the linearised Navier-Stokes operator. Thus, the core of the instability, also known in the literature as the wavemaker, is sought performing an analysis of the structural sensitivity of the unstable eigenmode to spatially localised feedbacks. The region located 15° downstream of the bend inlet, on the outer wall, is the most receptive to this kind of perturbations, and thus corresponds to where the instability originates. Since this region coincides with the outer-wall separation bubble, it is concluded that the instability is linked to the strong shear by the backflow phenomena. The present results are relevant for technical applications where bent pipes are frequently used, and their stability properties have hitherto not been studied.     

A mesoscopic modelling approach for direct numerical simulations of transition to turbulence in hypersonic flow with transpiration cooling

A rescaling methodology is developed for high-fidelity, cost-efficient direct numerical simulations (DNS) of flow through porous media, modelled at mesoscopic scale, in a hypersonic freestream. The simulations consider a Mach 5 hypersonic flow over a flat plate with coolant injection from a porous layer with 42 % porosity. The porous layer is designed using a configuration studied in the literature, consisting of a staggered arrangement of cylinder/sphere elements. A characteristic Reynolds number ${\mathit{Re}}_c$ of the flow in a pore cell unit is first used to impose aerodynamic similarity between different porous layers with the same porosity, $∊$, but different pore size. A relation between the pressure drop and the Reynolds number is derived to allow a controlled rescaling of the pore size from the realistic micrometre scales to higher and more affordable scales. Results of simulations carried out for higher cylinder diameters, namely 24 μm, 48 μm and 96 μm, demonstrate that an equivalent Darcy-Forchheimer behaviour to the reference experimental microstructure is obtained at the different pore sizes. The approach of a porous layer with staggered spheres is applied to a 3D domain case of porous injection in the Darcy limit over a flat plate, to study the transition mechanism and the associated cooling performance, in comparison with a reference case of slot injection. Results of the direct numerical simulations show that porous injection in an unstable boundary layer leads to a more rapid transition process, compared to slot injection. On the other hand, the mixing of coolant within the boundary layer is enhanced in the porous injection case, both in the immediate outer region of the porous layer and in the turbulent region. This has the beneficial effect of increasing the cooling performance by reducing the temperature near the wall, which provides a higher cooling effectiveness, compared to the slot injection case, even with an earlier transition to turbulence.     

Resolvent-based design and experimental testing of porous materials for passive turbulence control

An extended version of the resolvent formulation is used to evaluate the use of anisotropic porous materials as passive flow control devices for turbulent channel flow. The effect of these porous substrates is introduced into the governing equations via a generalized version of Darcy’s law. Model predictions show that materials with high streamwise permeability and low wall-normal permeability (${\varphi }_{\mathit{xy}}=k_{\mathit{xx}}/k_{\mathit{yy}}\gg 1$) can suppress resolvent modes resembling the energetic near-wall cycle. Based on these predictions, two anisotropic porous substrates with ${\varphi }_{\mathit{xy}}>1$ and ${\varphi }_{\mathit{xy}}<1$ were designed and fabricated for experiments in a benchtop water channel experiment. Particle Image Velocimetry (PIV) measurements were used to compute mean turbulence statistics and to educe coherent structure via snapshot Proper Orthogonal Decomposition (POD). Friction velocity estimates based on the Reynolds shear stress profiles do not show evidence of discernible friction reduction (or increase) over the streamwise-preferential substrate with ${\varphi }_{\mathit{xy}}>1$ relative to a smooth wall flow at identical bulk Reynolds number. A significant increase in friction is observed over the substrate with ${\varphi }_{\mathit{xy}}<1$. This increase in friction is linked to the emergence of spanwise rollers resembling Kelvin–Helmholtz vortices. Coherent structures extracted via POD analysis show qualitative agreement with model predictions.     

Effect of radius of toroidal square duct on the transition of electromagnetically driven liquid metal flow

Toroidal magnetohydrodynamic flows are important for fusion technology because of the blanket surrounding the core of the fusion machine. The effect of the radius of toroidal square ducts on flow transition of an electromagnetically driven liquid metal is assessed. The flow governing equations are solved numerically using direct numerical simulation. Critical Reynolds (Re) numbers where transition occurs are obtained for Hartmann (Ha) numbers in the range of 3 < Ha < 500. The results are compared with previous experimental and numerical results of toroidal square ducts of various radii. It is shown that, independent of the toroidal duct mean radius (r), the transition occurs in three regimes: in the sub-magnetohydrodynamic regime for very low Ha numbers, in a plateau regime for 10 < Ha < 20 and in a higher Ha regime. By increasing the radius of the torus, a transition occurs at higher Reynolds numbers as turbulent fluctuations are developing inside the boundary layers of the cylindrical side walls. The results indicate that transition occurs at $\frac{Re}{\sqrt{r}}H̃a^{-3.39}$ in the sub-magnetohydrodynamic regime and at $\frac{Re}{\sqrt{r}}H̃a^{-1.4}$ in the magnetohydrodynamic one. The boundary layer thickness at the Hartmann walls were found to scale as ${\mathit{Ha}}^{-0.8}$.     

Hairpin vortices in the largest scale of turbulent boundary layers

We have conducted direct numerical simulations of a turbulent boundary layer for the momentum-thickness-based Reynolds number ${\mathit{Re}}_{\theta }$ = 180–4600. To extract the largest-scale vortices, we coarse-grain the fluctuating velocity fields by using a Gaussian filter with the filter width comparable to the boundary layer thickness. Most of the largest-scale vortices identified by isosurfaces of the second invariant of the coarse-grained velocity gradient tensor are similar to coherent vortices observed in low-Reynolds-number regions, that is, hairpin vortices or quasi-streamwise vortices inclined to the wall. We also develop a percolation analysis to investigate the threshold-dependence of the isosurfaces and objectively identify the largest-scale hairpin vortices in terms of the coarse-grained vorticity, which leads to the quantitative evidence that they never disappear even in fully developed turbulent regions. Hence, we conclude that hairpin vortices exist in the largest-scale structures irrespective of the Reynolds number.     

2D numerical contributions for the study of non-cohesive sediment transport beneath tidal bores

2D numerical simulations of tidal bores were obtained using the OpenFOAM CFD software to solve the Navier–Stokes equations by means of the Finite Volume Method by applying a LES turbulence model. The trajectories of non-cohesive sediment particles beneath tidal bores were estimated using a tracker method. Using the fourth order Runge–Kutta scheme, the tracker method solves the Maxey and Riley equations, which requires the knowledge of the velocity field at time t. From 2D numerical simulations of tidal bores, we proposed a classification of tidal bores with respect to the Froude number Fr (or r the ratio of water depths). For a Froude number $1<Fr<1.43$ ($1<r<1.57$), the tidal bore is undular. For a Froude number $1.43<Fr<1.57$ ($1.57<r<1.75$), the tidal bore is partially breaking, which is similar to the transitional tidal bore defined by Furgerot (2014). And for a Froude number $Fr>1.57$ ($r>1.75$), the tidal bore is totally breaking. The numerical results of trajectories of non-cohesive sediment particles are similar to the type of trajectories given by the analytical model proposed by Chen et al. (2012) with some modifications to take into account the effects of gravity, elevation, and attenuation. The parameters of modified Chen's model, ${\beta }_1$, ${\beta }_2$ and ${\beta }_3$, are linearly proportional to the Froude number Fr. This is because the level of turbulence for undular tidal bores is low. The flow induced by an undular tidal bore is not complex. This physical phenomenon is quasi linear. The parameter ${\beta }_1$, related to the front celerity of the undular tidal bore, decreases when the Froude number Fr increases. The parameter ${\beta }_2$, related to the elevation, increases when the Froude number Fr increases. And the parameter ${\beta }_3$, related to the attenuation of the secondary waves, increases when the Froude number Fr increases.