The dimension of attractors underlying periodic turbulent Poiseuille flow

1992 ◽  
Vol 242 ◽  
pp. 1-29 ◽  
Author(s):  
Laurence Keefe ◽  
Parviz Moin ◽  
John Kim
Keyword(s):  
Reynolds Number ◽  
Lower Bound ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Coarse Grained ◽  
Turbulent Shear ◽  
Global Dynamics ◽  
Computational Domain ◽  
Turbulent Shear Flow ◽  
Attractor Dimension

Using a coarse grained (16 × 33 × 8) numerical simulation, a lower bound on the Lyapunov dimension, Dλ, of the attractor underlying turbulent, periodic Poiseuille flow at a pressure-gradient Reynolds number of 3200 has been calculated to be approximately 352. These results were obtained on a spatial domain with streamwise and spanwise periods of 1.6π, and correspond to a wall-unit Reynolds number of 80. Comparison of Lyapunov exponent spectra from this and a higher-resolution (16 × 33 × 16) simulation on the same domain shows these spectra to have a universal shape when properly scaled. Using these scaling properties, and a partial exponent spectrum from a still higher-resolution (32 × 33 × 32) simulation, we argue that the actual dimension of the attractor underlying motion on the given computational domain is approximately 780. The medium resolution calculation establishes this dimension as a strong lower bound on this computational domain, while the partial exponent spectrum calculated at highest resolution provides some evidence that the attractor dimension in fully resolved turbulence is unlikely to be substantially larger. These calculations suggest that this periodic turbulent shear flow is deterministic chaos, and that a strange attractor does underly solutions to the Navier–Stokes equations in such flows. However, the magnitude of the dimension measured invalidates any notion that the global dynamics of such turbulence can be attributed to the interaction of a few degrees of freedom. Dynamical systems theory has provided the first measurement of the complexity of fully developed turbulence; the answer has been found to be dauntingly high.

Effect of the shear parameter on velocity-gradient statistics in homogeneous turbulent shear flow

2011 ◽  
Vol 678 ◽  
pp. 14-40 ◽  
Author(s):  
JUAN C. ISAZA ◽  
LANCE R. COLLINS
Keyword(s):  
Reynolds Number ◽  
Shear Flow ◽  
Scaling Laws ◽  
Stokes Equations ◽  
Strain Tensor ◽  
Turbulent Shear ◽  
Small Scale ◽  
Turbulent Shear Flow ◽  
Velocity Statistics ◽  
Shear Parameter

The effect of the shear parameter on the small-scale velocity statistics in an homogeneous turbulent shear flow is investigated using direct numerical simulations (DNSs) of the incompressible Navier–Stokes equations on a 5123 grid. We use a novel pseudo-spectral algorithm that allows us to set the initial value of the shear parameter in the range 3–30 without the shortcomings of previous numerical approaches. We find that the tails of the probability distribution function of components of the vorticity vector and rate-of-strain tensor are progressively distorted with increasing shear parameter. Furthermore, we show that the shear parameter has a direct effect on the structure of the vorticity field, which manifests through changes in its alignment with the eigenvectors of the rate-of-strain tensor. We also find that increasing the shear parameter causes the main contribution to enstrophy production to shift from the nonlinear terms to the rapid terms (terms that are proportional to the mean shear) due to the aforementioned changes in the alignment. We attempt to explain these trends using viscous rapid distortion theory; however, while the theory does capture some effects of the shear parameter, it fails to predict the correct dependence on Reynolds number. Comparisons with recent experiments are also shown. The trends predicted by the DNS and the experiments are in good agreement. Moreover, the prefactors in the Reynolds number scaling laws for the skewness and flatness of the longitudinal velocity derivative are shown to have a statistically significant dependence on the shear parameter.

Computations of Ship Stern and Wake Flow and Comparisons with Experiment

1990 ◽  
Vol 34 (03) ◽  
pp. 179-193
Author(s):  
V. C. Patel ◽  
H. C. Chen ◽  
S. Ju
Keyword(s):  
Experimental Data ◽  
Shear Flow ◽  
Numerical Method ◽  
Stokes Equations ◽  
Ship Hull ◽  
Wake Flow ◽  
Navier Stokes ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Navier Stokes Equations

A numerical method for the solution of the Reynolds-averaged Navier-Stokes equations has been employed to study the turbulent shear flow over the stern and in the wake of a ship hull. Detailed comparisons are made between the numerical results and available experimental data to show that most of the important overall features of such flows can now be predicted with considerable accuracy.

Stationary perturbations of Couette–Poiseuille flow: the flow development in long cavities and channels

1995 ◽  
Vol 292 ◽  
pp. 153-182 ◽  
Author(s):  
Jennifer R. Stocker ◽  
Peter W. Duck
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Branching Processes ◽  
Reynolds Numbers ◽  
Streamwise Direction ◽  
End Effects ◽  
Driven Cavity ◽  
Streamwise Pressure Gradient

We consider stationary perturbations to Couette–Poiseuille flows. These may be considered to be related to far downstream/upstream entry/end effects in flow inside long cavities and channels. Three distinct classes of basic flow are considered, all of which are exact solutions of the Navier–Stokes equations. We first study the problem in the case of Poiseuille flow, and are able to explain a previous discrepancy between fully numerical results, and asymptotic theory valid for large Reynolds numbers, R. The second case, which may be derived from a combination of an imposed streamwise pressure gradient and sliding of the upper channel wall, is for the particular situation where the flow on the lower surface is on the verge of reversing direction. The third case is relevant to the flow inside a long driven cavity (with closed ends, no imposed streamwise pressure gradient and no net mass flux). The flow is driven exclusively by a sliding top wall and mass conservation demands that the flow is no longer unidirectional.For low Reynolds numbers, the stationary eigenvalues in all cases considered are complex (and hence are not monotonic in the streamwise direction). Indeed as R → 0 the eigenvalues become completely independent of the base profile. As the Reynolds number is increased, the eigenvalues generally undergo a number of branching processes switching between being complex and real (and vice versa) in nature, and at large Reynolds numbers fall broadly into three distinct categories, namely O(1), O(R−1/7) and O(1/R). In this limit the eigenvalues may be either complex or real (tending to monotonic eigensolutions in the streamwise direction).Of particular interest are certain of the O(1) eigensolutions for the ‘driven-cavity’ problem, in the high-Reynolds-number limit; these turn out to be highly oscillatory (WKB-type) over much of the cavity section.In all three cases, we use a combination of numerical and asymptotic techniques, and a thorough comparison between results thus obtained is made.

scholarly journals Characterization of the Behavior of Confined Laminar Round Jets

2012 ◽  
Author(s):  
D. Tyler Landfried ◽  
A. Jana ◽  
M. L. Kimber
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Flow Characteristics ◽  
Expansion Ratio ◽  
Downstream Location ◽  
Practical Applications ◽  
Free Jet ◽  
Laminar Jet ◽  
Jet Behavior

Confined laminar fluid jets have many practical applications in industry. Several examples include expansions in pipes and flow of gas into a large plenum. While much consideration has been given experimentally to heat transfer and pressure gradients within the confinement, little attention has been paid to quantify the velocity profiles and transitions between various flow behaviours. Using a finite volume CFD code, OpenFOAM ®, the Navier-Stokes equations were solved for varying expansion ratio, 1/ε = renclosure/rj, and varying Reynolds numbers. In the present analysis, Reynolds number based on the inlet jet diameter is varied from 30 to 70, well within the accepted range for laminar jet behavior. The expansion ratio, 1/ε is varied from 20–200. Of primary focus in the current study are compact correlations for the jet centreline velocity as a function of jet Reynolds number, Rej and expansion ratio. Similar functional dependences for the “linear” decay region of the jet, and the location of the stagnation point on the enclosure wall, are also investigated. These are all important features of the global flow field for the confined jet. Results suggest that initially, the flow characteristics are identical to a free jet. At some downstream location, the presence of the enclosure is felt by the jet and deviations begin to be seen from free jet behavior. This transition region continues until at a sufficiently large downstream location, the flow becomes fully developed, internal Poiseuille flow. In this paper, we analyse these transition regions and offer explanations and practical correlations to successfully predict the important flow physics that occur between free jet behavior and Poiseuille flow. Key dimensionless parameters are identified, the magnitude of which can be used to classify the flow conditions.

scholarly journals Variations on Kolmogorov flow: turbulent energy dissipation and mean flow profiles

2011 ◽  
Vol 670 ◽  
pp. 204-213 ◽  
Author(s):  
B. ROLLIN ◽  
Y. DUBIEF ◽  
C. R. DOERING
Keyword(s):  
Reynolds Number ◽  
Stokes Flow ◽  
Three Dimensional ◽  
Turbulent Energy ◽  
Dissipation Factor ◽  
Turbulent Shear ◽  
Flow Profile ◽  
Turbulent Shear Flow ◽  
Mean Flow ◽  
Flow Profiles

The relation between the form of a body force driving a turbulent shear flow and the dissipation factor β = ϵℓ/U3 is investigated by means of rigorous upper bound analysis and direct numerical simulation. We consider unidirectional steady forcing functions in a three-dimensional periodic domain and observe that a rigorous infinite Reynolds number bound on β displays the same qualitative behaviour as the computationally measured dissipation factor at finite Reynolds number as the force profile is varied. We also compare the measured mean flow profiles with the Stokes flow profile for the same forcing. The mean and Stokes flow profiles are strikingly similar at the Reynolds numbers obtained in the numerical simulations, lending quantitative credence to the notion of a turbulent eddy viscosity.

Direct Numerical Simulation of Homogeneous Turbulent Shear Flow Containing Bubbles

2006 ◽  
Author(s):  
T. Kawamura ◽  
T. Nakatani
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Production Rate ◽  
Turbulent Flows ◽  
Reynolds Numbers ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Hypothetical Model ◽  
Turbulent Reynolds Number

Direct numerical simulations of homogeneous shear turbulent flows containing deformable bubbles were carried out for clarifying the mechanism of drag reduction by microbubbles. The results show that presence of bubbles can suppress or enhance the development of turbulence depending on condition. The dissipation rate of turbulent kinetic energy is always increased by bubbles, while the production rate can be either increased or decreased depending on the turbulent and shear Reynolds numbers. As a result, the growth rate of turbulent kinetic energy can be either increased or decreased by bubbles depending on conditions. It was shown that the production rate tends to decrease at smaller shear Reynolds number, at larger turbulent Reynolds number, and at larger Weber number. Based on the results, a hypothetical model to explain the dependency on the Reynolds numbers has been proposed.

Bounds for turbulent shear flow

1970 ◽  
Vol 41 (1) ◽  
pp. 219-240 ◽  
Author(s):  
F. H. Busse
Keyword(s):  
Shear Flow ◽  
Turbulent Flow ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Navier Stokes ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Navier Stokes Equations ◽  
Experimental Values

Bounds on the transport of momentum in turbulent shear flow are derived by variational methods. In particular, variational problems for the turbulent regimes of plane Couette flow, channel flow, and pipe flow are considered. The Euler equations resemble the basic Navier–Stokes equations of motion in many respects and may serve as model equations for turbulence. Moreover, the comparison of the upper bound with the experimental values of turbulent momentum transport shows a rather close similarity. The same fact holds with respect to other properties when the observed turbulent flow is compared with the structure of the extremalizing solution of the variational problem. It is suggested that the instability of the sublayer adjacent to the walls is responsible for the tendency of the physically realized turbulent flow to approach the properties of the extremalizing vector field.

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