Laminar to Turbulence Transitions

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the
field of fluid mechanics. This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is
a three-dimensional phenomenon.

The subgrid-scale (SGS) model in a large-eddy simulation (LES) generally operates on a range of scales that is marginally resolved by discretization schemes. Consequently, the discretization scheme’s truncation error and the subgrid-scale model are linked, which raises the question of how accurate the computational results are. The link between the SGS model and truncation error can be beneficially exploited by developing discretization methods for subgrid-scale modeling, or vice versa. Approaches where the SGS model and the numerical discretization scheme are fully merged are called implicit LES (ILES) methods.
In order to improve on modeling uncertainties, a systematic framework is proposed for design, analysis, and optimization of nonlinear discretization schemes for implicit LES. The resulting adaptive local deconvolution method (ALDM) for implicit LES is a finite volume method based on a nonlinear deconvolution operator and a numerical flux function. Free parameters inherent to the discretization allow to control the truncation error. They are calibrated in such a way that the truncation error acts as a physically motivated SGS model. An automatic optimization based on an evolutionary algorithm is employed to obtain a set of parameters that results in an optimum match between the spectral numerical viscosity and theoretical predictions of the spectral eddy viscosity for isotropic turbulence. The method is formulated for LES of turbulent flows governed by the incompressible Navier-Stokes equations and for passive-scalar mixing.
ALDM has shown the potential for providing a reliable, accurate, and efficient method for LES. Various applications, such as three-dimensional homogeneous isotropic turbulence, transitional and turbulent plane channel flow, and turbulent boundary-layer separation, demonstrate the good performance of the implicit model. Computational results agree well with theory and experimental data and show that the implicit SGS model performs at least as well as established explicit models, for most considered applications the performance is even better. This is possible because physical reasoning is incorporated into the design of the discretization scheme and discretization effects are fully taken into account within the SGS model formulation.

""This paper provides the analysis of bursting and transition to turbulence in a Couette flow, based on the growth of nonlinear optimal disturbances. We use a global variational procedure to identify such optimal disturbances, defined as those initial perturbations yielding the largest energy growth at a given target time, for given Reynolds number and initial energy. The nonlinear optimal disturbances are found to be characterized by a basic structure, composed by inclined streamwise vortices along localized regions of low and high momentum. Such a basic structure closely recalls the one found in the boundary layer flow (Cherubini et al. (2011)), indicating that this structure may be considered the most ”energetic” one at short target times. However, small differences in the shape of these optimal perturbations, due to different levels of the initial energy or target
time assigned in the optimization process, may produce remarkable differences in their evolution towards turbulence. In particular, DNSs have shown that optimal disturbances
obtained for large initial energies and target times induce bursting events, whereas for lower values of these parameters the flow is directly attracted towards the turbulent state. For this reason, the optimal disturbances have been classified into two classes, the highly dissipative and the short-path perturbations. Both classes lead the flow to turbulence skipping the phases of streaks formation and secondary instability which are
typical of the classical transition scenario for shear flows. The dynamics of this transition scenario exploits three main features of the non-linear optimal disturbances: i) the large initial value of the streamwise velocity component; ii) the streamwise dependence of the disturbance; iii) the presence of initial inclined streamwise vortices. The short-path perturbations are found to spend a considerable amount of time in the vicinity of the
edge state (Schneider (2008)), whereas the highly dissipative optimal disturbances pass closer to the edge, but they are rapidly repelled away from it, leading the flow to high values of the dissipation rate. After this dissipation peak, the trajectories do not lead towards the turbulent attractor, but they spend some time in the vicinity of an unstable periodic orbit (UPO). This behaviour led us to conjecture that bursting events can be obtained not only as homoclinic orbits approaching the UPO, as recently found by van
Veen & Kawahara (2011), but also as heteroclinic orbits between the equilibrium solution on the edge and the UPO.""

"The understanding of transition in shear flows has recently progressed along new paradigms based on the central role of coherent flow structures and their nonlinear interactions. We follow such paradigms to identify, by means of a non-linear
optimization of the energy growth at short time, the initial perturbation which most easily induces transition in a boundary layer. Moreover, a bisection procedure has been used to identify localized flow structures living on the edge of chaos, found to be
populated by hairpin vortices and streaks. Such an edge structure appears to act as a relative attractor for the trajectory of the laminar base state perturbed by the initial finite amplitude disturbances, mediating the route to turbulence of the flow, via the triggering of a regeneration cycle of  and hairpin structures at different space and time scales. These findings introduce a new, purely non-linear scenario of transition in a boundary-layer flow."

This contribution presents a stability analysis for compressible boundary layer flows over indented surfaces. Specifically, the effects of increasing depth D /δ * and Ma  number on perturbation time-decay rates and spatial amplification factors are quantified and compared with those of an unindented configuration. The indented surfaces represent aeronautical lifting surfaces endowed with the smooth gap resulting when a filler material applied at the junction of leading-edge and wing-box components retracts upon its curing process. Since the configuration considered is such that the parallel/weakly-parallel assumptions are necessarily compromised, a global temporal stability analysis is considered in this study. Our analysis does not require a parallel flow constrain, and hence it is believed to be valid when two dimensional effects are relevant. We find that small surface modifications enhance certain flow instabilities. An increase in Ma ∞ enhances further this behaviour: for the D /δ * = 1.5, Ma  = 0.5 case, amplification factors at a given location can be up to 20 times larger than those corresponding to the unindented case.

The flow characteristics in the inertial Reynolds number regime are investigated in a mono-dispersed random pack porous media. Time-resolved particle image velocimetry (PIV) is used to visualize the velocity field in a low aspect ratio bed with 15 mm glass beads. An aqueous solution of Ammunium Thiocynante is used as the working fluid to facilitate matching the solid-fluid refractiveindices. In order to illuminate the inertial regime characteristics, two pore Reynolds number of 100 and 270 are examined. Also, due to the random nature of the packing several pore geometries are compared to identify local scaling used to define the inertial regime effects. Discrete vortical flow structures are evaluated using LES (lowpass filtering) decomposition, in conjunction with criticalpoint analysis of the local velocity gradient tensor. The identified scales associated with the vortical elements are compared based on Reynolds number and pore geometry. Implementing circulation as an integral measure of all vortical structures locally at the pore-scale level demonstrated a linear attitude over the range of Reynolds numbers. Evolution of inertial effects within pore-regions are indicated to be the primary driving mechanism for the emergence of swirling structures passing through the PIV field of view at the onset of turbulence. KEYWORDS Particle image velocimetry (PIV), vortex identification, vorti-cal structure, transition to turbulence, porous media, inertial flow regime, random packed bed, refractive index matched (RIM/MIR) NOMENCLATURE d p [µm] Diameter of PIV tracer particles β [mm] Characteristic length scale of fluid phase n L Liquid refractive index n S Solid refractive index D B [m] Bead diameter D H [m] Hydraulic diameter D tracer [m] Tracer particle diameter L x [mm] Test section width x-direction L y [mm] Test section height y-direction Re p Pore Reynolds number Re P local Local Pore Reynolds number V Darcy [mm/s] Superficial or Darcy velocity V int [m/s] Interstitial velocity Γ 1 [m 2 /s] Vortex center identification scheme Γ [m 2 /s] Circulation λ [nm] Wavelength of light λ ci [1/s] Swirl strength µ [Kg/ms] Dynamic viscosity of fluid phase ν [m 2 /s] Kinematic viscosity of fluid phase φ Porosity ρ [Kg/m 3 ] Density of fluid phase ρ p [gr/cm 3 ] Density of PIV tracer particles rms Root mean square St p The stokes number of PIV tracer particles 1

The motion of fluids which are incompressible could be described by the Navier-Stokes differential equations. However, the
three-dimensional Navier-Stokes equations for modelling turbulence misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would greatly affect the field of fluid mechanics. In this paper, which had been published in an international journal in 2010, a reasoned, practical approach towards resolving the issue is adopted and a practical, statistical kind of mathematical solution is proposed.

Flight pressure and heat flux data have been compared to angle-of-attack-and yaw-dependent computational-fluid-dynamics results for pressure distribution as well as laminar and turbulent heat-transfer results at three time points in the ascent trajectory. Computed pressures, normalized by freestream pressure, were interpolated to the flight Mach numbers at each time point throughout the ascent and descent trajectories, and angle of attack and yaw were estimated from measured pressure by determining the combination minimizing the difference between the measured and computed pressures. The resulting vehicle attitude was compared to the vehicle attitude derived from inertial measurement unit results from the flight. The two methods showed excellent agreement for the entirety of the ascent and reentry portions of the trajectory. A similar normalization of the laminar and turbulent computational-fluid-dynamics heat transfer results into Stanton number distributions was compared to flight heat transfer measurements, and transition times at a given location were inferred. Computational heat conduction analysis verified assumptions in the calculation of heat flux from temperature. Nomenclature c p = specific heat capacity, constant pressure, J∕…kg ⋅ k† M = Mach number p = pressure, Pa _ q = heat flux, W∕m 2 Re = Reynolds number St = Stanton number T = temperature, K t = time, s u = velocity, m∕s x = streamwise distance, m α = angle of attack, deg β = yaw, deg ρ = density, kg∕m 3 ϕ = angular location on vehicle surface, deg Subscripts CFD = derived from computational solution CL = centerline e = boundary-layer edge conditions F = derived from flight measurement FFT = fast Fourier transform IMU = recorded by the inertial measurement unit LE = leading edge RMS = derived from root-mean-squared error minimization between computational fluid dynamics and flight tr = at transition onset w = wall conditions 0 = stagnation conditions ∞ = freestream conditions

A series of slender-body hypervelocity boundary-layer instability and transition experiments were performed in the Caltech T5 Reflected Shock Tunnel. During this campaign, it became clear that the condition of the T5 shock tube would significantly affect the consistency of the instability and transition measurements; a regimen of cleaning was iterated on until satisfactory repeatability was achieved. In this work, a description of the cleaning regimen is given. Additionally, boundary-layer instability measurements and a statistical analysis of the boundary-layer transition scatter are presented for experiments before and after cleaning regimen implementation.

A dynamical system description of the transition process in shear flows with no linear instability starts with knowledge of exact coherent solutions, among them traveling waves (TWs) and relative periodic orbits (RPOs). We describe a numerical method to find such solutions in pipe flow and apply it in the vicinity of a Hopf bifurcation from a TW which looks to be especially relevant for transition. The dominant structural feature of the RPO solution is the presence of weakly modulated streaks. This RPO, like the TW from which it bifurcates, sits on the laminar-turbulent boundary separating initial conditions which lead to turbulence from those which immediately relaminarize.

"Flows over airfoils and blades in rotating machinery, for unmanned and micro-aerial vehicles, wind turbines, and propellers consist of a laminar boundary layer near the leading edge that is often followed by a laminar separation bubble and transition to turbulence further downstream. Typical RANS turbulence models are inadequate for such flows. Direct numerical simulation (DNS) is the most reliable, but is also the most computationally expensive alternative. This work assesses the capability of Immersed Boundary (IB) methods and Large Eddy Simulations (LES) to reduce the computational requirements for such flows and still provide high quality results. Two-dimensional and three-dimensional simulations of a laminar separation bubble on a NACA-0012 airfoil at Re = 50 000 at 5 degrees of incidence have been performed with an IB code and a commercial code using body fitted grids. Several Subgrid Scale (SGS) models have been implemented in both codes and their performance evaluated. For the two-dimensional simulations with the IB method the results show good agreement with DNS benchmark data for the pressure coefficient Cp and the friction coefficient Cf but only when using dissipative numerical schemes. There is evidence that this behavior can be attributed to the ability of dissipative schemes to damp numerical noise coming from the IB. For the three-dimensional simulations the results show a good prediction of the separation point, but inaccurate prediction of the reattachment point unless full DNS resolution is used. The commercial code shows good agreement with the DNS benchmark data in both two and three-dimensional simulations, but the presence of significant, unquantified numerical dissipation prevents a conclusive assessment of the actual prediction capabilities of very coarse LES with low order schemes in general case."

We propose a general strategy for determining the minimal finite amplitude disturbance to trigger transition to turbulence in shear flows. This involves constructing a variational problem that searches over all disturbances of fixed initial amplitude, which respect the boundary conditions, incompressibility and the Navier--Stokes equations, to maximise a chosen functional over an asymptotically long time period. The functional must be selected such that it identifies turbulent velocity fields by taking significantly enhanced values compared to those for laminar fields. We illustrate this approach using the ratio of the final to initial perturbation kinetic energies (energy growth) as the functional and the energy norm to measure amplitudes in the context of pipe flow. Our results indicate that the variational problem yields a smooth converged solution providing the amplitude is below the threshold amplitude for transition. This optimal is the nonlinear analogue of the well-studied (linear) transient growth optimal. At and above this threshold, the optimising search naturally seeks out disturbances that trigger turbulence by the end of the period, and convergence is then practically impossible. The first disturbance found to trigger turbulence as the amplitude is increased identifies the `minimal seed' for the given geometry and forcing (Reynolds number). We conjecture that it may be possible to select a functional such that the converged optimal below threshold smoothly converges to the minimal seed at threshold.  This seems at least approximately true for our choice of energy growth functional and the pipe flow geometry chosen here.

Linear transient growth analysis is commonly used to suggest the structure of disturbances which are particularly efficient in triggering transition to turbulence in shear flows. We demonstrate that the addition of nonlinearity to the analysis can substantially change the prediction made in pipe flow from simple two-dimensional streamwise rolls to a spanwise and cross-stream localized three-dimensional state. This new nonlinear optimal is demonstrably more efficient in triggering turbulence than the linear optimal indicating that there are better ways to design perturbations to achieve transition.

Transition to turbulence in randomly arranged porous media is observed in nature and industrial applications. The flow characteristics of these flows during transition are not well identified. This work describes the parameters influencing on overall mixing during the transition process from the perspective of scale of vortical structures and dispersion characteristics by addressing the following questions: (a) what are the dominant mechanisms evolution of scale of vortices, and (b) how does the inertial effects of vortical structures enhance the flow transport properties through tortuosity and dispersion. Time-resolved PIV is used to investigate the flow in the macro-scale Reynolds numbers from 100 to 1000 to show the pore- versus macro-scale effects on the scale of the flow dispersion, and their contribution in interpreting the overall flow mixing. Lagrangian mixing characteristics based on Eulerian local pore velocity variances is used to demonstrate the bed characteristics for flow in randomly distributed porous media flows. The dispersion asymptotically approaches 0.085 % of VintDH longitudinally which shows the turbulent transport is increased by enhancing the Reynolds number that matches very well with the literature.

Crossflow instability of a three-dimensional boundary layer is a common cause of transition in swept-wing flows. The boundary-layer flow modified by the presence of finite-amplitude crossflow modes is susceptible to high-frequency secondary instabil-ities, which are believed to ...