Stochastic low-dimensional modelling of a random laminar wake past a circular cylinder

2008 ◽  
Vol 606 ◽  
pp. 339-367 ◽  
Author(s):  
DANIELE VENTURI ◽  
XIAOLIANG WAN ◽  
GEORGE EM KARNIADAKIS
Keyword(s):  
Circular Cylinder ◽  
Turbulent Flows ◽  
Stokes Equations ◽  
Stochastic Simulations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Spatial Modes ◽  
Laminar Wake ◽  
Low Dimensional ◽  
Proper Orthogonal

We present a new compact expansion of a random flow field into stochastic spatial modes, hence extending the proper orthogonal decomposition (POD) to noisy (non-coherent) flows. As a prototype problem, we consider unsteady laminar flow past a circular cylinder subject to random inflow characterized as a stationary Gaussian process. We first obtain random snapshots from full stochastic simulations (based on polynomial chaos representations), and subsequently extract a small number of deterministic modes and corresponding stochastic modes by solving a temporal eigenvalue problem. Finally, we determine optimal sets of random projections for the stochastic Navier–Stokes equations, and construct reduced-order stochastic Galerkin models. We show that the number of stochastic modes required in the reconstruction does not directly depend on the dimensionality of the flow system. The framework we propose is general and it may also be useful in analysing turbulent flows, e.g. in quantifying the statistics of energy exchange between coherent modes.

scholarly journals Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

2009 ◽  
Vol 629 ◽  
pp. 41-72 ◽  
Author(s):  
ALEXANDER HAY ◽  
JEFFREY T. BORGGAARD ◽  
DOMINIQUE PELLETIER
Keyword(s):  
Sensitivity Analysis ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Selection Step ◽  
Low Dimensional ◽  
Proper Orthogonal

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

Computation of Vortex Lock-In in the Laminar Wake of a Circular Cylinder Using Unsteady Monopole Sources

1995 ◽  
Vol 117 (2) ◽  
pp. 227-233 ◽  
Author(s):  
Dartzi Pan ◽  
Yu-Chi Chin ◽  
Chih-Hao Chang
Keyword(s):  
Circular Cylinder ◽  
Flow Field ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Finite Volume Scheme ◽  
Cylinder Surface ◽  
Navier Stokes Equations ◽  
Laminar Wake ◽  
Lock In ◽  
Flux Difference

The vortex lock-in in the laminar wake behind a circular cylinder induced by the unsteady monopole source is numerically simulated in this paper. The artificial compressibility method is employed to solve the incompressible Navier-Stokes equations. A high-order accurate upwind flux-difference finite-volume scheme is used to discretize the flow field. The unsteady monopole source is simulated by a pulsating volume flux through the cylinder surface at a prescribed forcing frequency and amplitude. The forcing amplitude is set to a fixed value while the frequency is varied to search for the lock-in region. The flow field of the periodic lock-in state is examined in detail. Finally, the effects of a higher amplitude and a different source location are briefly investigated.

Numerical Simulation and Stability Analysis of Laminar Wake Behind a Circular Cylinder

1998 ◽  
Vol 14 (4) ◽  
pp. 183-192
Author(s):  
Mei-Jiau Huang
Keyword(s):  
Circular Cylinder ◽  
Stokes Equations ◽  
Maximum Growth ◽  
Spectral Element ◽  
Navier Stokes ◽  
Stream Velocity ◽  
Navier Stokes Equations ◽  
Splitting Technique ◽  
Growth Criterion ◽  
Laminar Wake

ABSTRACTThe transition of a viscous incompressible laminar flow behind a circular cylinder from a steady state to its wake, counterpart at a Reynolds number Re = 100, based on the free stream velocity (U) and the cylinder diameter (D), is directly simulated. The two-dimensional unsteady Navier-Stokes equations are solved numerically by taking advantage of the splitting technique and the spectral element method. The main goal of this work is to explore the frequency-selection mechanism of the wake through the use of the absolute/convective instability theory, which in turn is performed by investigating the one-dimensional Orr-Sommerfeld equation. It is found that the predicted onset frequency based on the maximum-growth criterion is in a good agreement with the numerically observed one, although the measured growth rate is found smaller. The saturated frequencies predicted by the maximum-growth criterion and Kock's transition criterion are the same and also close to the measured one. More simulation or experimental data are needed for a further conclusion however.

On Direct Numerical Simulation of Turbulent Flows

2011 ◽  
Vol 64 (2) ◽  
Author(s):  
Giancarlo Alfonsi
Keyword(s):  
Numerical Simulation ◽  
Direct Numerical Simulation ◽  
High Performance Computing ◽  
Turbulent Flows ◽  
High Performance ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Turbulent Shear ◽  
Navier Stokes Equations ◽  
Performance Computing

The direct numerical simulation of turbulence (DNS) has become a method of outmost importance for the investigation of turbulence physics, and its relevance is constantly growing due to the increasing popularity of high-performance-computing techniques. In the present work, the DNS approach is discussed mainly with regard to turbulent shear flows of incompressible fluids with constant properties. A body of literature is reviewed, dealing with the numerical integration of the Navier-Stokes equations, results obtained from the simulations, and appropriate use of the numerical databases for a better understanding of turbulence physics. Overall, it appears that high-performance computing is the only way to advance in turbulence research through the front of the direct numerical simulation.

Numerical solution of the early stage of the unsteady viscous flow around a circular cylinder: a comparison with experimental visualization and measurements

1985 ◽  
Vol 160 ◽  
pp. 93-117 ◽  
Author(s):  
Ta Phuoc Loc ◽  
R. Bouard
Keyword(s):  
Circular Cylinder ◽  
Flow Structure ◽  
Stokes Equations ◽  
Early Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Grid Systems ◽  
Unsteady Viscous Flow ◽  
Vorticity Transport ◽  
New Phenomena

Early stages of unsteady viscous flows around a circular cylinder at Reynolds numbers of 3 × 103 and 9.5 × 103 are analysed numerically by direct integration of the Navier–Stokes equations – a fourth-order finite-difference scheme is used for the resolution of the stream-function equation and a second-order one for the vorticity-transport equation. Evolution with time of the flow structure is studied in detail. Some new phenomena are revealed and confirmed by experiments.The influence of the grid systems and the downstream boundary conditions on the flow structure and the velocity profiles is reported. The computed results are compared qualitatively and quantitatively with experimental visualization and measurements. The comparison is found to be satisfactory.

CFD Analysis of Vortex Shedding in a Circular Cylinder: Flow Induced Vibration Design

2006 ◽  
Author(s):  
Nadeem Ahmed Sheikh ◽  
M. Afzaal Malik ◽  
Arshad Hussain Qureshi ◽  
M. Anwar Khan ◽  
Shahab Khushnood
Keyword(s):  
Circular Cylinder ◽  
Vortex Shedding ◽  
Stokes Equations ◽  
Boundary Layer Separation ◽  
The Body ◽  
Navier Stokes ◽  
Structural Vibrations ◽  
Flow Induced Vibration ◽  
Navier Stokes Equations ◽  
Implicit Approach

Flow past a blunt body, such as a circular cylinder, usually experiences boundary layer separation and very strong flow oscillations in the wake region behind the body at a discrete frequency that is correlated to the Reynolds number of the flow. The periodic nature of the vortex shedding phenomenon can sometimes lead to unwanted structural vibrations. The effect of vibrating instability of a single cylinder is investigated in a uniform flow using the power of computational methods. Fluid structure coupling procedure predicts the fluid forces responsible for structural vibrations. An implicit approach to the solution of the unsteady two-dimensional Navier-Stokes equations is used for computation of flow parameters. Calculations are performed in parallel using a domain re-meshing/deforming technique with efficient communication requirements. Results for the unsteady shedding flow behind a circular cylinder are presented with experimental comparisons, showing the feasibility of accurate, efficient, time-dependent estimation of shedding frequency and resulting vibrations.

Heat Transfer From a Circular Cylinder at Low Reynolds Numbers

1998 ◽  
Vol 120 (1) ◽  
pp. 72-75 ◽  
Author(s):  
V. N. Kurdyumov ◽  
E. Ferna´ndez
Keyword(s):  
Circular Cylinder ◽  
Energy Equation ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Order Unity ◽  
Navier Stokes Equations ◽  
Wide Range ◽  
Prandtl Numbers ◽  
High Degree

A correlation formula, Nu = W0(Re)Pr1/3 + W1(Re), that is valid in a wide range of Reynolds and Prandtl numbers has been developed based on the asymptotic expansion for Pr → ∞ for the forced heat convection from a circular cylinder. For large Prandtl numbers, the boundary layer theory for the energy equation is applied and compared with the numerical solutions of the full Navier Stokes equations for the flow field and energy equation. It is shown that the two-terms asymptotic approximation can be used to calculate the Nusselt number even for Prandtl numbers of order unity to a high degree of accuracy. The formulas for coefficients W0 and W1, are provided.

scholarly journals The emergence of localized vortex–wave interaction states in plane Couette flow

2013 ◽  
Vol 721 ◽  
pp. 58-85 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall ◽  
Andrew Walton
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
High Reynolds Number ◽  
Stokes Equations ◽  
Wave Interaction ◽  
Critical Layer ◽  
Navier Stokes ◽  
Turbulent Spots ◽  
Navier Stokes Equations ◽  
High Reynolds

AbstractThe recently understood relationship between high-Reynolds-number vortex–wave interaction theory and computationally generated self-sustaining processes provides a possible route to an understanding of some of the underlying structures of fully turbulent flows. Here vortex–wave interaction (VWI) theory is used in the long streamwise wavelength limit to continue the development found at order-one wavelengths by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). The asymptotic description given reduces the Navier–Stokes equations to the so-called boundary-region equations, for which we find equilibrium states describing the change in the VWI as the wavelength of the wave increases from $O(h)$ to $O(Rh)$, where $R$ is the Reynolds number and $2h$ is the depth of the channel. The reduced equations do not include the streamwise pressure gradient of the perturbation or the effect of streamwise diffusion of the wave–vortex states. The solutions we calculate have an asymptotic error proportional to ${R}^{- 2} $ when compared to the full Navier–Stokes equations. The results found correspond to the minimum drag configuration for VWI states and might therefore be of relevance to the control of turbulent flows. The key feature of the new states discussed here is the thickening of the critical layer structure associated with the wave part of the flow to completely fill the channel, so that the roll part of the flow is driven throughout the flow rather than as in Hall & Sherwin as a stress discontinuity across the critical layer. We identify a critical streamwise wavenumber scaling, which, when approached, causes the flow to localize and take on similarities with computationally generated or experimentally observed turbulent spots. In effect, the identification of this critical wavenumber for a given value of the assumed high Reynolds number fixes a minimum box length necessary for the emergence of localized structures. Whereas nonlinear equilibrium states of the Navier–Stokes equations are thought to form a backbone on which turbulent flows hang, our results suggest that the localized states found here might play a related role for turbulent spots.

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