scholarly journals The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160585 ◽  
Author(s):  
Divya Venkataraman ◽  
Samriddhi Sankar Ray
Keyword(s):  
Energy Spectrum ◽  
Numerical Simulations ◽  
Euler Equations ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Infinite Dimensional ◽  
Localized Structures ◽  
Finite Dimensional ◽  
Incompressible Euler

Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber K G ), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localized structures, called tygers (Ray et al. 2011 Phys. Rev. E 84 , 016301 ( doi:10.1103/PhysRevE.84.016301 )), which eventually lead to completely thermalized states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time τ c at which thermalization is triggered and show that τ c ∼ K G ξ , with ξ = − 4 9 . Our results have several implications, including for the analyticity strip method, to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.

scholarly journals Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence

2008 ◽  
Vol 602 ◽  
pp. 175-207 ◽  
Author(s):  
L. E. JONES ◽  
R. D. SANDBERG ◽  
N. D. SANDHAM
Keyword(s):  
Numerical Simulations ◽  
Reverse Flow ◽  
Three Dimensional ◽  
Transition Process ◽  
Direct Numerical Simulations ◽  
Transition To Turbulence ◽  
Absolute Instability ◽  
Laminar Separation ◽  
Velocity Magnitude ◽  
Separation Bubbles

Direct numerical simulations (DNS) of laminar separation bubbles on a NACA-0012 airfoil at Rec=5×104 and incidence 5° are presented. Initially volume forcing is introduced in order to promote transition to turbulence. After obtaining sufficient data from this forced case, the explicitly added disturbances are removed and the simulation run further. With no forcing the turbulence is observed to self-sustain, with increased turbulence intensity in the reattachment region. A comparison of the forced and unforced cases shows that the forcing improves the aerodynamic performance whilst requiring little energy input. Classical linear stability analysis is performed upon the time-averaged flow field; however no absolute instability is observed that could explain the presence of self-sustaining turbulence. Finally, a series of simplified DNS are presented that illustrate a three-dimensional absolute instability of the two-dimensional vortex shedding that occurs naturally. Three-dimensional perturbations are amplified in the braid region of developing vortices, and subsequently convected upstream by local regions of reverse flow, within which the upstream velocity magnitude greatly exceeds that of the time-average. The perturbations are convected into the braid region of the next developing vortex, where they are amplified further, hence the cycle repeats with increasing amplitude. The fact that this transition process is independent of upstream disturbances has implications for modelling separation bubbles.

Direct Numerical Simulations of Transitional Separation–Bubble Development in Swept–Blade Flow Conditions

2012 ◽  
Author(s):  
Joshua R. Brinkerhoff ◽  
Metin I. Yaras
Keyword(s):  
Boundary Layer ◽  
Numerical Simulations ◽  
Shear Layer ◽  
Pressure Field ◽  
Separation Bubble ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Small Scale ◽  
Flow Conditions ◽  
Crossflow Instability

This paper describes numerical simulations of the instability mechanisms in a separation bubble subjected to a three-dimensional freestream pressure distribution. Two direct numerical simulations are performed of a separation bubble with laminar separation and turbulent reattachment under low freestream turbulence at flow Reynolds numbers and streamwise pressure distributions that approximate the conditions encountered on the suction side of typical low-pressure gas-turbine blades with blade sweep angles of 0° and 45°. The three-dimensional pressure field in the swept configuration produces a crossflow-velocity component in the laminar boundary layer upstream of the separation point that is unstable to a crossflow instability mode. The simulation results show that crossflow instability does not play a role in the development of the boundary layer upstream of separation. An increase in the amplification rate and most amplified disturbance frequency is observed in the separated-flow region of the swept configuration, and is attributed to boundary-layer conditions at the point of separation that are modified by the spanwise pressure gradient. This results in a slight upstream movement of the location where the shear layer breaks down to small-scale turbulence and modifies the turbulent mixing of the separated shear layer to yield a downstream shift in the time-averaged reattachment location. The results demonstrate that although crossflow instability does not appear to have a noticeable effect on the development of the transitional separation bubble, the 3D pressure field does indirectly alter the separation-bubble development by modifying the flow conditions at separation.

scholarly journals Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Thierry Horsin ◽  
Mohamed Ali Jendoubi
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Equilibrium Points ◽  
Infinite Dimensional ◽  
Infinite Dimensional Dynamical Systems ◽  
Finite Dimensional ◽  
Angle Condition ◽  
Non Local ◽  
Second Order Systems

<p style='text-indent:20px;'>In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.</p>

Self-Organization in Three-Dimensional Hydrodynamic Turbulence Self-Organization in Three-Dimensional Hydrodynamic Turbulence

1990 ◽  
Vol 45 (9-10) ◽  
pp. 1059-1073 ◽  
Author(s):  
G. Knorr ◽  
J. P. Lynov ◽  
H. L. Pécseli
Keyword(s):  
Euler Equations ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Self Organization ◽  
Two Dimensional ◽  
Hydrodynamic Turbulence ◽  
Available Energy ◽  
Wave Numbers ◽  
Negative Helicity ◽  
Incompressible Euler

Abstract The three-dimensional incompressible Euler equations are expanded in eigenflows of the curl operator, which represent positive and negative helicity flows in a particularly simple and convenient way. Four different basic types of interactions between eigenflows are found. Two represent an "inverse cascade", the interaction familiar from the two-dimensional Euler equations, in which only modes of the same sign of the helicity interact. The other two interactions mix positive and negative helicity modes. Only these interactions can transport all of the available energy to higher wave numbers. Initial conditions, which lead to the appearance of structures and self-organization, are discussed.

scholarly journals Comparison of Flame Propagation Statistics Extracted from Direct Numerical Simulation Based on Simple and Detailed Chemistry—Part 1: Fundamental Flame Turbulence Interaction

Energies ◽  
2021 ◽  
Vol 14 (17) ◽  
pp. 5548
Author(s):  
Felix Benjamin Keil ◽  
Marvin Amzehnhoff ◽  
Umair Ahmed ◽  
Nilanjan Chakraborty ◽  
Markus Klein
Keyword(s):  
Numerical Simulations ◽  
Flame Propagation ◽  
High Performance ◽  
Three Dimensional ◽  
Quantitative Agreement ◽  
Combustion Regime ◽  
Computing Methods ◽  
Direct Numerical Simulations ◽  
Tangential Strain ◽  
Detailed Chemistry

In the present study, flame propagation statistics from turbulent statistically planar premixed flames obtained from simple and detailed chemistry, three-dimensional Direct Numerical Simulations, were evaluated and compared to each other. To this end, a new database was established encompassing five different conditions on the turbulent premixed combustion regime diagram, using nearly identical numerical methods and the same initial and boundary conditions. A detailed discussion of the advantages and limitations of both approaches is provided, including the difference in carbon footprint for establishing the database. It is shown that displacement speed statistics and their interrelation with curvature and tangential strain rate are in very good qualitative and reasonably good quantitative agreement between simple and detailed chemistry Direct Numerical Simulations. Hence, it is concluded that simple chemistry simulations should retain their importance for future combustion research, and the environmental impact of high-performance computing methods should be carefully chosen in relation to the goals to be achieved.

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