Nonlinear Direct Numerical Simulations Research Articles

Bifurcation analysis of double cavity flows

The first few bifurcations in a two-dimensional incompressible double cavity flow are investigated using the linear stability analysis, the Floquet analysis, and the nonlinear direct numerical simulations (DNS). The prediction of the critical Reynolds number and the type of bifurcation (Hopf, pitchfork, inverse pitchfork, and Neimark–Sacker), which depend on cavity configuration, by the linear stability analysis and the Floquet analysis is consistent with nonlinear DNS. The nonlinear DNS results show that the state of the system passes through multiple intermediate (unstable) states before it reaches the stable attractor (heteroclinic chain), and the type of intermediate states depends on initial conditions. The intermediate states are reported as the asymptotic state in the literature for some flow conditions because it is not known a priori how long it will take to reach the asymptotic state in nonlinear simulations. The present study reports the actual asymptotic state for those flow conditions.

From thin plates to Ahmed bodies: linear and weakly nonlinear stability of rectangular prisms

We study the stability of laminar wakes past three-dimensional rectangular prisms. The width-to-height ratio is set to $W/H=1.2$ , while the length-to-height ratio $1/6< L/H<3$ covers a wide range of geometries from thin plates to elongated Ahmed bodies. First, global linear stability analysis yields a series of pitchfork and Hopf bifurcations: (i) at lower Reynolds numbers $Re$ , two stationary modes, $A$ and $B$ , become unstable, breaking the top/bottom and left/right planar symmetries, respectively; (ii) at larger $Re$ , two oscillatory modes become unstable and, again, each mode breaks one of the two symmetries. The critical $Re$ values of these four modes increase with $L/H$ , reproducing qualitatively the trend of stationary and oscillatory bifurcations in axisymmetric wakes (e.g. thin disk, sphere and bullet-shaped bodies). Next, a weakly nonlinear analysis based on the two stationary modes $A$ and $B$ yields coupled amplitude equations. For Ahmed bodies, as $Re$ increases, state $(A,0)$ appears first, followed by state $(0,B)$ . While there is a range of bistability of those two states, only $(0,B)$ remains stable at larger $Re$ , similar to the static wake deflection (across the larger base dimension) observed in the turbulent regime. The bifurcation sequence, including bistability and hysteresis, is validated with fully nonlinear direct numerical simulations, and is shown to be robust to variations in $W$ and $L$ in the range of common Ahmed bodies.

Onset of vortex shedding around a short cylinder

This paper presents results of three-dimensional direct numerical simulations (DNS) and global linear stability analyses of a viscous incompressible flow past a finite-length cylinder with two free flat ends. The cylindrical axis is normal to the streamwise direction. The work focuses on the effects of aspect ratios (in the range of $0.5\leq {\small \text{AR}} \leq 2$ , cylinder length over diameter) and Reynolds numbers ( $Re\leq 1000$ based on cylinder diameter and uniform incoming velocity) on the onset of vortex shedding in this flow. All important flow patterns have been identified and studied, especially as ${\small \text{AR}}$ changes. The appearance of a steady wake pattern when ${\small \text{AR}} \leq 1.75$ has not been discussed earlier in the literature for this flow. Linear stability analyses based on the time-mean flow has been applied to understand the Hopf bifurcation past which vortex shedding happens. The nonlinear DNS results indicate that there are two vortex shedding patterns at different $Re$ , one is transient and the other is nonlinearly saturated. The vortex-shedding frequencies of these two flow patterns correspond to the eigenfrequencies of the two global modes in the stability analysis of the time-mean flow. Wherever possible, we compare the results of our analyses to those of the flows past other short- ${\small \text{AR}}$ bluff bodies in order that our discussions bear more general meanings.

Structured input–output analysis of transitional wall-bounded flows

Input–output analysis of transitional channel flows has proven to be a valuable analytical tool for identifying important flow structures and energetic motions. The traditional approach abstracts the nonlinear terms as forcing that is unstructured, in the sense that this forcing is not directly tied to the underlying nonlinearity in the dynamics. This paper instead employs a structured-singular-value-based approach that preserves certain input–output properties of the nonlinear forcing function in an effort to recover the larger range of key flow features identified through nonlinear analysis, experiments and direct numerical simulation (DNS) of transitional channel flows. Application of this method to transitional plane Couette and plane Poiseuille flows leads to not only the identification of the streamwise coherent structures predicted through traditional input–output approaches, but also the characterization of the oblique flow structures as those requiring the least energy to induce transition, in agreement with DNS studies, and nonlinear optimal perturbation analysis. The proposed approach also captures the recently observed oblique turbulent bands that have been linked to transition in experiments and DNS with very large channel size. The ability to identify the larger amplification of the streamwise varying structures predicted from DNS and nonlinear analysis in both flow regimes suggests that the structured approach allows one to maintain the nonlinear effects associated with weakening of the lift-up mechanism, which is known to dominate the linear operator. Capturing this key nonlinear effect enables the prediction of a wider range of known transitional flow structures within the analytical input–output modelling paradigm.

Numerical investigations of fully nonlinear water waves generated by moving bottom topography

This paper is concerned with propagation of water waves induced by moving bodies with uniform velocity on the bottom of a channel, a simple model for prescribed underwater landslides. The fluid is assumed to be inviscid and incompressible, and the flow, irrotational. We apply this model to a variety of test problems, and particular attention is paid to long-time dynamics of waves induced by two landslide bodies moving with the same speed. We focus on the transcritical regime where the linear theory fails to depict the wave phenomena even in the qualitative sense since it predicts an infinite growth in amplitude. In order to resolve this problem, weakly nonlinear theory or direct numerical simulations for the fully nonlinear equations is required. Comparing results of the linear full-dispersion theory, the linear shallow water equations, the forced Korteweg-de Vries model, and the full Euler equations, we show that water waves generated by prescribed underwater landslides are characterized by the Froude number, sizes of landslide bodies and distance between them. Particularly, in the transcritical regime, the second body plays a key role in controlling the criticality for equal landslide bodies, while for unequal body heights, the higher one controls the criticality. The results obtained in the current paper complement numerical studies based on the forced Korteweg-de Vries equation and the nonlinear shallow water equations by Grimshaw and Maleewong (J. Fluid Mech. 2015, 2016).

On the stability of a Blasius boundary layer subject to localised suction

In this study the origins of premature transition due to oversuction in boundary layers are studied. An infinite row of circular suction pipes that are mounted at right angles to a flat plate subject to a Blasius boundary layer is considered. The interaction between the flow originating from neighbouring holes is weak and for the parameters investigated, the pipe is always found to be unsteady regardless of the state of the flow in the boundary layer. A stability analysis reveals that the appearance of boundary layer transition can be associated with a linear instability in the form of two unstable eigenmodes inside the pipe that have weak tails, which extend into the boundary layer. Through an energy budget and a structural sensitivity analysis, the origin of this flow instability is traced to the structures developing inside the pipe near the pipe junction. Although the amplitudes of the modes in the boundary layer are orders of magnitude smaller than the corresponding amplitudes inside the pipe, a Koopman analysis of the data gathered from a nonlinear direct numerical simulation confirms that it is precisely these disturbances that are responsible for transition to turbulence in the boundary layer due to oversuction.

Joint instability and abrupt nonlinear transitions in a differentially rotating plasma

Global magnetohydrodynamic (MHD) instabilities are investigated in a computationally tractable two-dimensional model of the solar tachocline. The model’s differential rotation yields stability in the absence of a magnetic field, but if a magnetic field is present, a joint instability is observed. We analyse the nonlinear development of the instability via fully nonlinear direct numerical simulation, the generalized quasi-linear approximation (GQL) and direct statistical simulation (DSS) based upon low-order expansion in equal-time cumulants. As the magnetic diffusivity is decreased, the nonlinear development of the instability becomes more complicated until eventually a set of parameters is identified that produces a previously unidentified long-term cycle in which energy is transformed from kinetic energy to magnetic energy and back. We find that the periodic transitions, which mimic some aspects of solar variability – for example, the quasiperiodic seasonal exchange of energy between toroidal field and waves or eddies – are unable to be reproduced when eddy-scattering processes are excluded from the model.

Vortex pairing in jets as a global Floquet instability: modal and transient dynamics

The spontaneous pairing of rolled-up vortices in a laminar jet is investigated as a global secondary instability of a time-periodic spatially developing vortex street. The growth of subharmonic perturbations, associated with vortex pairing, is analysed both in terms of modal Floquet instability and in terms of transient growth dynamics. The article has the double objective to outline a toolset for the global analysis of time-periodic flows, and to leverage such an analysis for a fresh view on the vortex pairing phenomenon. Axisymmetric direct numerical simulations (DNS) of jets with single-frequency inflow forcing are performed, in order to identify combinations of the Reynolds and Strouhal numbers for which vortex pairing is naturally observed. The same DNS calculations are then repeated with an added time-delay control term, which artificially suppresses pairing, so as to obtain time-periodic unpaired base flows for linear stability analysis. It is demonstrated that the natural occurrence of vortex pairing in nonlinear DNS coincides with a linear subharmonic Floquet instability of the underlying unpaired vortex street. However, DNS results suggest that the onset of pairing involves much stronger temporal growth of subharmonic perturbations than that predicted by modal Floquet analysis, as well as a spatial distribution of these fast-growing perturbation structures that is inconsistent with the unstable Floquet mode. Singular value decomposition of the phase-shift operator (the operator that maps a given perturbation field to its state one flow period later) is performed for an analysis of optimal transient growth in the vortex street. Non-modal mechanisms near the jet inlet are thus found to provide a fast route towards the limit-cycle regime of established vortex pairing, in good agreement with DNS observations. It is concluded that modal Floquet analysis accurately predicts the parameter regime where sustained vortex pairing occurs, but that the bifurcation scenario under typical conditions is dominated by transient growth phenomena.

Слабо-нелинейный анализ устойчивости термоакустического адвективного течения

The influence of a traveling sound wave on the stability of an advective flow with respect to spiral perturbations is studied. A weak nonlinear stability analysis and direct numerical simulation were performed. From the condition of solvability of a third-order problem, a differential equation is obtained, which describes the evolution of amplitude functions. At the same time, to calculate the constants in this equation, it is necessary to determine the eigenfunctions of the linear problem of stability and the solution of some inhomogeneous linear problems. The shooting method was used using the orthogonalization procedure. It is shown that supercritical regimes appear soft and stable near the threshold with respect to modulation. Direct numerical simulation of the problem was carried out using the finite difference method. A study of convective flow with a significant distance from the stability threshold was carried out. It has been established that for a certain value of parameters in the layer, oscillations of a complex structure occur.

The linear instability of the stratified plane Couette flow

We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonal to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the streamwise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number $Re$ and the Froude number $Fr$, which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e. $Fr\sim 1$) and above a moderate value of the Reynolds number $Re\gtrsim 700$. The instability results from a wave resonance mechanism already known in the context of channel flows – for instance, unstratified plane Couette flow in the shallow-water approximation. The result is confirmed by fully nonlinear direct numerical simulations and, to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water, linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold values of $Fr$ and $Re$ indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although slightly modified due to streamwise confinement.

Linear and non-linear analyses on the onset of miscible viscous fingering in a porous medium

The onset of miscible viscous fingering in porous media was analyzed theoretically. The linear stability equations were derived in the self-similar domain, and solved through the modal and non-modal analyses. In the non-modal analysis, adjoint equations were derived using the Lagrangian multiplier technique. Through the non-modal analysis, we show that initially the system is unconditionally stable even in the unfavorable viscosity distribution, and there exists the most unstable initial disturbance. To relate the theoretical predictions with the experimental work, nonlinear direct numerical simulations were also conducted. The present stability condition explains the system more reasonably than the previous results based on the conventional quasi-steady state approximation.

Shock-induced energy transfers in dense gases

Dense gases are characterised by molecules featuring large numbers of active degrees of freedom (quantified by the cv/R ratio). The isentropes in such gases have the distinct property of following rather closely the isotherms (the two become identical in the limit of cv/R going to infinity). Near the liquid-vapour critical point, this makes the isentropes very shallow and possibly concave (in the pressure-specific volume diagram). Whilst shallow isentropes are desirable when designing expanders (i.e. a large specific-volume increase may be achieved for virtually no pressure drop), could such extreme compressibility effects modify turbulence in a profound manner? This paper discusses two particularly interesting aspects: (i) shock-refraction properties (i.e. the way a shock can redistribute the energy of incoming perturbations), (ii) enstrophy production in homogeneous turbulence. A linear interaction analysis (LIA) is conducted on the shock configuration for which the incoming perturbation is decomposed into linear modes of the compressible Euler equations. The transmission coefficients relative to each eigen modes are solved analytically and results are compared against fully non-linear compressible direct numerical simulation reproducing the weak perturbation of an isolated two-dimensional compression shock wave. The linear analysis is found to be capable of predicting the shock-induced redistribution of the energy of the incoming perturbation between the different eigen modes. Non-ideal gas effects are observed both analytically and numerically with especially an unusual selective response for some particular choice of incoming Mach number. A two-dimensional isotropic turbulence configuration is then numerically investigated for the case of an inviscid compressible dense-gas flow close to the liquid-vapour critical point. Strong non-ideal-gas effects on enstrophy production are observed with the formation of eddy shocklets. In both cases non-convex isentropes close to the liquid-vapour critical point are extremely influential in letting both the shock and the turbulence redistribute any supply of turbulence kinetic energy in ways which are simply not observable in ideal gases. This will hopefully spark enthusiasm amongst turbulence modellers (and their end users?).

Generalised quasilinear approximation of the helical magnetorotational instability

Motivated by recent advances in direct statistical simulation (DSS) of astrophysical phenomena such as out-of-equilibrium jets, we perform a direct numerical simulation (DNS) of the helical magnetorotational instability (HMRI) under the generalised quasilinear approximation (GQL). This approximation generalises the quasilinear approximation (QL) to include the self-consistent interaction of large-scale modes, interpolating between fully nonlinear DNS and QL DNS whilst still remaining formally linear in the small scales. In this paper we address whether GQL can more accurately describe low-order statistics of axisymmetric HMRI when compared with QL by performing DNS under various degrees of GQL approximation. We utilise various diagnostics, such as energy spectra in addition to first and second cumulants, for calculations performed for a range of Reynolds and Hartmann numbers (describing rotation and imposed magnetic field strength respectively). We find that GQL performs significantly better than QL in describing the statistics of the HMRI even when relatively few large-scale modes are kept in the formalism. We conclude that DSS based on GQL (GCE2) will be significantly more accurate than that based on QL (CE2).

Mechanisms of flow tripping by discrete roughness elements in a swept-wing boundary layer

The effects of a spanwise row of finite-size cylindrical roughness elements in a laminar, compressible, three-dimensional boundary layer on a wing profile are investigated by direct numerical simulations (DNS). Large elements are capable of immediately tripping turbulent flow by either a strong, purely convective or an absolute/global instability in the near wake. First we focus on an understanding of the steady near-field past a finite-size roughness element in the swept-wing flow, comparing it to a respective case in unswept flow. Then, the mechanisms leading to immediate turbulence tripping are elaborated by gradually increasing the roughness height and varying the disturbance background level. The quasi-critical roughness Reynolds number above which turbulence sets in rapidly is found to be $Re_{kk,qcrit}\approx 560$ and global instability is found only for values well above 600 using nonlinear DNS; therefore the values do not differ significantly from two-dimensional boundary layers if the full velocity vector at the roughness height is taken to build $Re_{kk}$. A detailed simulation study of elements in the critical range indicates a changeover from a purely convective to a global instability near the critical height. Finally, we perform a three-dimensional global stability analysis of the flow field to gain insight into the early stages of the temporal disturbance growth in the quasi-critical and over-critical cases, starting from a steady state enforced by damping of unsteady disturbances.

Passivity-based output-feedback control of turbulent channel flow

This paper describes a robust linear time-invariant output-feedback control strategy to reduce turbulent fluctuations, and therefore skin-friction drag, in wall-bounded turbulent fluid flows, that nonetheless gives performance guarantees in the nonlinear turbulent regime. The novel strategy is effective in reducing the supply of available energy to feed the turbulent fluctuations, expressed as reducing a bound on the supply rate to a quadratic storage function. The nonlinearity present in the equations that govern the dynamics of the flow is known to be passive and can be considered as a feedback forcing to the linearised dynamics (a Lur’e decomposition). Therefore, one is only required to control the linear dynamics in order to make the system close to passive. The ten most energy-producing spatial modes of a turbulent channel flow were identified. Passivity-based controllers were then generated to control these modes. The controllers require measurements of streamwise and spanwise wall-shear stress, and they actuate via wall transpiration. Nonlinear direct numerical simulations demonstrated that these controllers were capable of significantly reducing the turbulent energy and skin-friction drag of the flow.

Modal instability of the flow in a toroidal pipe

The modal instability encountered by the incompressible flow inside a toroidal pipe is studied, for the first time, by means of linear stability analysis and direct numerical simulation (DNS). In addition to the unquestionable aesthetic appeal, the torus represents the smallest departure from the canonical straight pipe flow, at least for low curvatures. The flow is governed by only two parameters: the Reynolds number $\mathit{Re}$ and the curvature of the torus ${\it\delta}$, i.e. the ratio between pipe radius and torus radius. The absence of additional features, such as torsion in the case of a helical pipe, allows us to isolate the effect that the curvature has on the onset of the instability. Results show that the flow is linearly unstable for all curvatures investigated between 0.002 and unity, and undergoes a Hopf bifurcation at $\mathit{Re}$ of about 4000. The bifurcation is followed by the onset of a periodic regime, characterised by travelling waves with wavelength $\mathit{O}(1)$ pipe diameters. The neutral curve associated with the instability is traced in parameter space by means of a novel continuation algorithm. Tracking the bifurcation provides a complete description of the modal onset of instability as a function of the two governing parameters, and allows a precise calculation of the critical values of $\mathit{Re}$ and ${\it\delta}$. Several different modes are found, with differing properties and eigenfunction shapes. Some eigenmodes are observed to belong to groups with a set of common characteristics, deemed ‘families’, while others appear as ‘isolated’. Comparison with nonlinear DNS shows excellent agreement, confirming every aspect of the linear analysis, its accuracy, and proving its significance for the nonlinear flow. Experimental data from the literature are also shown to be in considerable agreement with the present results.

Soret-driven convection and separation of binary mixtures in a horizontal porous cavity submitted to cross heat fluxes

An analytical and numerical study of Soret-driven convection in a horizontal porous layer saturated by a binary fluid and subjected to uniform cross heat fluxes is presented. The flow is driven by the combined buoyancy effect due to temperature and induced mass fraction variations through a binary water ethanol mixture. In the first part of the study, a closed-form analytical solution in the limit of a large aspect ratio of the cell (A >> 1) is developed. We are mainly concerned with the determination of the mass fraction gradient of the component of interest along the horizontal direction, which determines the species separation. In the second part, numerous numerical simulations are carried out in order to validate the analytical results and extend heat and mass transfer to an area not covered by the analytical study. Good agreement is found between analytical and numerical results concerning the species separation obtained for a unicellular flow. In this configuration, the Soret separation process is improved by two control parameters: the heat flux density imposed on the horizontal walls of the cell and the ratio, a, of heat flux density imposed on vertical walls to that on horizontal walls. The influence of the heat flux density ratio, a, on the transient regime (relaxation time) is also investigated numerically. We observe that an increase in the parameter a leads to a decrease in the relaxation time. Thus, for a cell heated from below without lateral heating, the onset of convection from the mechanical equilibrium state is analyzed. The linear stability analysis shows that the equilibrium solution loses its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio and the normalized porosity of the medium. The linear stability results are widely corroborated by direct 2D numerical simulations. The thresholds of various multicellular solutions are determined in terms of the governing parameters of the problem using nonlinear direct numerical simulations.

Evolution of the single-mode Rayleigh-Taylor instability under the influence of time-dependent accelerations.

From nonlinear models and direct numerical simulations we report on several findings of relevance to the single-mode Rayleigh-Taylor (RT) instability driven by time-varying acceleration histories. The incompressible, direct numerical simulations (DNSs) were performed in two (2D) and three dimensions (3D), and at a range of density ratios of the fluid combinations (characterized by the Atwood number). We investigated several acceleration histories, including acceleration profiles of the general form g(t)∼t^{n}, with n≥0 and acceleration histories reminiscent of the linear electric motor experiments. For the 2D flow, results from numerical simulations compare well with a 2D potential flow model and solutions to a drag-buoyancy model reported as part of this work. When the simulations are extended to three dimensions, bubble and spike growth rates are in agreement with the so-called level 2 and level 3 models of Mikaelian [K. O. Mikaelian, Phys. Rev. E 79, 065303(R) (2009)10.1103/PhysRevE.79.065303], and with corresponding 3D drag-buoyancy model solutions derived in this article. Our generalization of the RT problem to study variable g(t) affords us the opportunity to investigate the appropriate scaling for bubble and spike amplitudes under these conditions. We consider two candidates, the displacement Z and width s^{2}, but find the appropriate scaling is dependent on the density ratios between the fluids-at low density ratios, bubble and spike amplitudes are explained by both s^{2} and Z, while at large density differences the displacement collapses the spike data. Finally, for all the acceleration profiles studied here, spikes enter a free-fall regime at lower Atwood numbers than predicted by all the models.

Lagrangian flows within reflecting internal waves at a horizontal free-slip surface

In this paper sequel to Zhou and Diamessis [“Reflection of an internal gravity wave beam off a horizontal free-slip surface,” Phys. Fluids 25, 036601 (2013)], we consider Lagrangian flows within nonlinear internal waves (IWs) reflecting off a horizontal free-slip rigid lid, the latter being a model of the ocean surface. The problem is approached both analytically using small-amplitude approximations and numerically by tracking Lagrangian fluid particles in direct numerical simulation (DNS) datasets of the Eulerian flow. Inviscid small-amplitude analyses for both plane IWs and IW beams (IWBs) show that Eulerian mean flow due to wave-wave interaction and wave-induced Stokes drift cancels each other out completely at the second order in wave steepness A, i.e., O(A2), implying zero Lagrangian mean flow up to that order. However, high-accuracy particle tracking in finite-Reynolds-number fully nonlinear DNS datasets from the work of Zhou and Diamessis suggests that the Euler-Stokes cancelation on O(A2) is not c...

Electrokinetic instability of liquid micro- and nanofilms with a mobile charge

The instability of ultra-thin films of an electrolyte bordering a dielectric gas in an external tangential electric field is scrutinized. The solid wall is assumed to be either a conducting or charged dielectric surface. The problem has a steady one-dimensional solution. The theoretical results for a plug-like velocity profile are successfully compared with available experimental data. The linear stability of the steady-state flow is investigated analytically and numerically. Asymptotic long-wave expansion has a triple-zero singularity for a dielectric wall and a quadruple-zero singularity for a conducting wall, and four (for a conducting wall) or three (for a charged dielectric wall) different eigenfunctions. For infinitely small wave numbers, these eigenfunctions have a clear physical meaning: perturbations of the film thickness, of the surface charge, of the bulk conductivity, and of the bulk charge. The numerical analysis provides an important result: the appearance of a strong short-wave instability. At increasing Debye numbers, the short-wave instability region becomes isolated and eventually disappears. For infinitely large Weber numbers, the long-wave instability disappears, while the short-wave instability persists. The linear stability analysis is complemented by a nonlinear direct numerical simulation. The perturbations evolve into coherent structures; for a relatively small external electric field, these are large-amplitude surface solitary pulses, while for a sufficiently strong electric field, these are short-wave inner coherent structures, which do not disturb the surface.