Minimal Flow Unit Research Articles

Using direct numerical simulation to analyze and improve hot-wire probe sensor and array configurations for simultaneous measurement of the velocity vector and the velocity gradient tensor

Multi-sensor, hot-wire probes of various configurations have been used for 25 years to simultaneously measure the velocity vector and the velocity gradient tensor in turbulent flows. This is the same period in which direct numerical simulations (DNS) were carried out to investigate these flows. Using the first DNS of a turbulent boundary layer, Moin and Spalart [“Contributions of numerical simulation data bases to the physics, modeling and measurement of turbulence,” NASA Technical Memorandum 100022 (1987)] examined, virtually, the performance of a two-sensor X-array probe with the sensors idealized as points in the numerical grid. Subsequently, several investigators have used DNS for similar studies. In this paper we use a highly resolved minimal channel flow DNS, following Jiménez and Moin [“The minimal flow unit in near-wall turbulence,” J. Fluid Mech. 225, 213 (1991)], to study the performance of an 11-sensor probe. Our previous studies of this type have indicated that, on balance, a probe of the design described here may provide the most accurate measurements of many of the statistics formed from the velocity vector and the velocity gradient tensor (rms and skewness values of the velocity and vorticity components as well as the Reynolds shear stress and the dissipation and production rates). The results of the present study show that, indeed, the sensor and array configurations of a probe of this design are considerably better than previous designs that have been used, and they are likely to give reasonably satisfactory results for such measurements with a real probe in a real bounded flow.

Computational approaches to aspect-ratio-dependent upper bounds and heat flux in porous medium convection

Direct numerical simulation (DNS) has shown that Rayleigh–Bénard convection in a fluid-saturated porous medium self-organizes into narrowly spaced plumes at (ostensibly) asymptotically high values of the Rayleigh number Ra . In this Letter a combination of DNS and upper bound theory is used to investigate the dependence of the Nusselt number Nu on the domain aspect ratio L at large Ra . A novel algorithm is introduced to solve the optimization problems arising from the upper bound analysis, allowing for the best available bounds to be extended up to Ra ≈ 2.65 × 10 4 . The dependence of the bounds on L ( Ra ) is explored and a “minimal flow unit” is identified. • DNS is used to quantify the aspect-ratio ( L ) dependency of Nu ( Ra ) in porous medium convection. • A “minimal flow unit” for porous medium convection is identified. • Rigorous upper bounds on Nu ( Ra ) are computed as a function of L for Ra ⩽ 2.65 × 10 4 . • A numerical scheme is developed for solving upper bound variational problems without continuation.

Intermittent dynamics of turbulence hibernation in Newtonian and viscoelastic minimal channel flows

AbstractMaximum drag reduction (MDR), the asymptotic upper limit of reduction in turbulent friction drag by polymer additives, is the most important unsolved problem in viscoelastic turbulence. Recent studies of turbulence in minimal flow units have identified time intervals showing key features of MDR. These intervals, denoted ‘hibernating turbulence’ are found in both Newtonian and viscoelastic flows. The present study provides a comprehensive examination of this turbulence hibernation phenomenon in the minimal channel geometry, and discusses its impact on the turbulent dynamics and drag reduction. Similarities between hibernating turbulence and MDR are established in terms of both flow statistics (an intermittency factor, mean and fluctuating components of velocity) and flow structure (weak vortices and nearly streamwise-invariant kinematics). Hibernation occurs more frequently at high levels of viscoelasticity, leading to flows that increasingly resemble MDR. Viscoelasticity facilitates the occurrence of hibernation by suppressing the conventional ‘active’ turbulence, but has little influence on hibernation itself. At low Weissenberg number $\mathit{Wi}$, the average duration of active turbulence intervals is constant, but above a critical value of $\mathit{Wi}$, the duration decreases dramatically, and accordingly, the fraction of time spent in hibernation increases. This observation can be explained with a simple mathematical model that posits that the lifetime of an active turbulence interval is the time that it takes for the turbulence to stretch polymer molecules to a certain threshold value; once the molecules exceed this threshold, they exert a large enough stress on the flow to suppress the active turbulence. This model predicts an explicit form for the duration as a function of $\mathit{Wi}$ and the simulation results match this prediction very closely. The critical point where hibernation frequency becomes substantially increased coincides with the point where qualitative changes are observed in overall flow statistics – the transition between ‘low-drag-reduction’ and ‘high-drag-reduction’ regimes. Probability density functions of important variables reveal a much higher level of intermittency in the turbulent dynamics after this transition. It is further confirmed that hibernating turbulence is a Newtonian structure during which polymer extension is small. Based on these results, a framework is proposed that explains key transitions in viscoelastic turbulence, especially the convergence toward MDR.

Turbulent drag reduction and multistage transitions in viscoelastic minimal flow units

The observation that addition of a minute amount of flexible polymers to fluid reduces turbulent friction drag is well known. However, many aspects of this drag reduction phenomenon are not well understood; in particular, the origin of the maximum drag reduction (MDR) asymptote, a universal upper limit on drag reduction by polymers, remains an open question. This study focuses on the drag reduction phenomenon in the plane Poiseuille geometry in a parameter regime close to the laminar–turbulent transition. By minimizing the size of the periodic simulation box to the lower limit for which turbulence persists, the essential self-sustaining turbulent motions are isolated. In these ‘minimal flow unit’ (MFU) solutions, a series of qualitatively different stages consistent with previous experiments is observed, including an MDR stage where the mean flow rate is found to be invariant with respect to changing polymer-related parameters. Before the MDR stage, an additional transition exists between a relatively low degree (LDR) and a high degree (HDR) of drag reduction. This transition occurs at about 13%–15% of drag reduction and is characterized by a sudden increase in the minimal box size, as well as many qualitative changes in flow statistics. The observation of LDR–HDR transition at less than 15% drag reduction shows for the first time that it is a qualitative transition instead of a quantitative effect of the amount of drag reduction. Spatio-temporal flow structures change substantially upon this transition, suggesting that two distinct types of self-sustaining turbulent dynamics are observed. In LDR, as in Newtonian turbulence, the self-sustaining process involves one low-speed streak and its surrounding streamwise vortices; after the LDR–HDR transition, multiple streaks are present in the self-sustaining structure and complex intermittent behaviour of the streaks is observed. This multistage scenario of LDR–HDR–MDR recovers all key transitions commonly observed and studied at much higher Reynolds numbers.

Low-dimensional models for turbulent plane Couette flow in a minimal flow unit

We model turbulent plane Couette flow in the minimal flow unit (MFU) – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence – by expanding the velocity field as a sum of optimal modes calculated via proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier–Stokes equations onto these modes. We first consider a 6-mode (11-dimensional) model and study the effects of including losses to neglected modes. Ignoring these, the model reproduces turbulent statistics acceptably, but fails to reproduce dynamics; including them, we find a stable periodic orbit that captures the regeneration cycle dynamics and agrees well with direct numerical simulations. However, restriction to as few as six modes artificially constrains the relative magnitudes of streamwise vortices and streaks and so cannot reproduce stability of the laminar state or properly account for bifurcations to turbulence as Reynolds number increases. To address this issue, we develop a second class of models based on ‘uncoupled’ eigenfunctions that allow independence among streamwise and cross-stream velocity components. A 9-mode (31-dimensional) model produces bifurcation diagrams for steady and periodic states in qualitative agreement with numerical Navier–Stokes solutions, while preserving the regeneration cycle dynamics. Together, the models provide empirical evidence that the ‘backbone’ for MFU turbulence is a periodic orbit, and support the roll–streak–breakdown–roll reformation picture of shear-driven turbulence.

Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial

The proper orthogonal decomposition identifies basis functions or modes which optimally capture the average energy content from numerical or experimental data. By projecting the Navier–Stokes equations onto these modes and truncating, one can obtain low-dimensional ordinary differential equation models for fluid flows. In this paper we present a tutorial on the construction of such models. In addition to providing a general overview of the procedure, we describe two different ways to numerically calculate the modes, show how symmetry considerations can be exploited to simplify and understand them, comment on how parameter variations are captured naturally in such models, and describe a generalization of the procedure involving projection onto uncoupled modes that allow streamwise and cross-stream components to evolve independently. We illustrate for the example of plane Couette flow in a minimal flow unit – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence.

Periodic Orbits and Chaotic Sets in a Low-Dimensional Model for Shear Flows

We consider the dynamics of a low-dimensional model for turbulent shear flows. The model is based on Fourier modes and describes sinusoidal shear flow, in which fluid between two free-slip walls experiences a sinusoidal body force. The model contains nine modes, most of which have a direct hydrodynamical interpretation. We analyze the stationary states and periodic orbits for the model for two different domain sizes. Several kinds of bifurcations are identified, including saddle-node bifurcations, a period doubling cascade, and Hopf bifurcations of the periodic orbits. For both domain sizes, long-lived transient chaos appears to be associated with the presence of a large number of unstable periodic orbits. For the smaller minimal flow unit domain, it is found that a periodic solution is stable over a range of Reynolds numbers, and its bifurcations lead to the existence of a chaotic attractor. The model illustrates many phenomena observed and speculated to exist in the transition to turbulence in linearly stable shear flows.

A periodic-like solution in channel flow

We search channel flow for unsteady solutions for different Reynolds numbers and configurations by extending a shooting method which was previously used to obtain a travelling-wave solution. A general initial condition is considered. A periodic-like solution to the incompressible Navier–Stokes equations in a minimal flow unit is found. One cycle of the solution consists of two typical intervals: a single-streak period and a double-streak period. The solution seems to be periodic; however, it cannot be distinguished from a heteroclinic cycle which consists of two heteroclinic orbits connecting two single-streak solutions, because the solution is tracked only for one and half periods.

An implicit velocity decoupling procedure for the incompressible Navier–Stokes equations

AbstractAn efficient numerical method to solve the unsteady incompressible Navier–Stokes equations is developed. A fully implicit time advancement is employed to avoid the Courant–Friedrichs–Lewy restriction, where the Crank–Nicolson discretization is used for both the diffusion and convection terms. Based on a block LU decomposition, velocity–pressure decoupling is achieved in conjunction with the approximate factorization. The main emphasis is placed on the additional decoupling of the intermediate velocity components with only nth time step velocity. The temporal second‐order accuracy is preserved with the approximate factorization without any modification of boundary conditions. Since the decoupled momentum equations are solved without iteration, the computational time is reduced significantly. The present decoupling method is validated by solving several test cases, in particular, the turbulent minimal channel flow unit. Copyright © 2002 John Wiley & Sons, Ltd.

Fully-Implicit Decoupling Method for Incompressible Navier-Stokes Equations

A new efficient numerical method for computing three-dimensional, unsteady, incompressible flows is presented. To eliminate the restriction of CFL condition, a fully-implicit time advancement in which the Crank-Nicolson method is used for both the diffusion and convection terms, is adopted. Based on an approximate block LU decomposition method, the velocity -pressure decoupling is achieved. The additional decoupling of the intermediate velocity components in the convection term is made for the fully -implicit time advancement scheme. Since the iterative procedures for the momentum equations are not required, the velocity components decouplings bring forth the reduction of computational cost. The second-order accuracy in time of the present numerical algorithm is ascertained by computing decaying vortices. The present decoupling method is applied to minimal channel flow unit with DNS (Direct Numerical Simulation).

A low-dimensional approach for the minimal flow unit

The proper orthogonal decomposition (POD) is applied to the minimal flow unit (MFU) of a turbulent channel flow. Our purpose is to establish a numerical validation of low-dimensional models based on the POD. The simplest (two-mode) model possible is built for the simplified flow in the minimal unit. The dynamical behaviour predicted by the model is compared with that actually occurring in the direct numerical simulation of the flow. The various modelling assumptions which underlie the construction of low-dimensional models are examined and confronted with numerical evidence. The relationship between intermittency in the MFU and intermittent low-dimensional parameters is investigated closely. The agreement observed is quite satisfactory, especially given the crudeness of the truncation considered. To further demonstrate the adequacy of the model, we develop a dynamical filtering procedure to recover information from realistic (partial) measurements. The success obtained illustrates the versatility of the low-dimensional paradigm.

Reconstructing the flow in the wall region from wall sensors

In this paper we examine how the flow in the wall region can be predicted from realistic (partial) measurements. The underlying motivation of this study is the improvement of control schemes for near-wall flows. We propose a method based on the proper orthogonal decomposition which provides estimated amplitudes for the coherent structures (i.e., the large scales) of the flow from wall measurements. The method is tested for the direct numerical simulation of a minimal flow unit. The large scales obtained by reconstruction from wall data are compared to those of the velocity field in the wall layer. The dominant structures—the streamwise streaks—are well recovered, the cross-stream motions less so since they are associated with higher-order structures unaccounted for in our truncation. We defined “rescaled” eigenfunctions to try to improve the representation of the cross-stream components of the flow. Aliasing effects due to realistic (large) sensor spacings were examined. We find that the spanwise spacing is the limiting factor for the estimation, so that a realistic grid will affect the reconstruction by at least 20% compared to full wall information.

Active control in the turbulent wall layer of a minimal flow unit

Direct simulations of flow in a channel with complex, time-dependent wall geometries facilitate an investigation of smart skin control in a turbulent wall layer (with skin friction drag reduction as the goal). The test bed is a minimal flow unit, containing one pair of coherent structures in the near-wall region: a high- and a low-speed streak. The controlling device consists of an actuator, Gaussian in shape and approximately twelve wall units in height, that emerges from one of the channel walls. Raising the actuator underneath a low-speed streak effects an increase in drag, raising it underneath a high-speed streak effects a reduction – indicating a mechanism for control. In the high-speed region, fast-moving fluid is lifted by the actuator away from the wall, allowing the adjacent low-speed region to expand and thereby lowering the average wall shear stress. Conversely, raising an actuator underneath a low-speed streak allows the adjacent high-speed region to expand, which increases skin drag.

The minimal flow unit in near-wall turbulence

Direct numerical simulations of unsteady channel flow were performed at low to moderate Reynolds numbers on computational boxes chosen small enough so that the flow consists of a doubly periodic (inxandz) array of identical structures. The goal is to isolate the basic flow unit, to study its morphology and dynamics, and to evaluate its contribution to turbulence in fully developed channels. For boxes wider than approximately 100 wall units in the spanwise direction, the flow is turbulent and the low-order turbulence statistics are in good agreement with experiments in the near-wall region. For a narrow range of widths below that threshold, the flow near only one wall remains turbulent, but its statistics are still in fairly good agreement with experimental data when scaled with the local wall stress. For narrower boxes only laminar solutions are found. In all cases, the elementary box contains a single low-velocity streak, consisting of a longitudinal strip on which a thin layer of spanwise vorticity is lifted away from the wall. A fundamental period of intermittency for the regeneration of turbulence is identified, and that process is observed to consist of the wrapping of the wall-layer vorticity around a single inclined longitudinal vortex.