arXiv subjects

Siegfried Grossmann

Publications and source records attributed to Siegfried Grossmann.

At least 37 records · Page 2Linked to original sources

arXiv2011

Horizontal Structures of Velocity and Temperature Boundary Layers in 2D Numerical Turbulent Rayleigh-Bénard Convection

We investigate the structures of the near-plate velocity and temperature profiles at different horizontal positions along the conducting bottom (and top) plate of a Rayleigh-Bénard convection cell, using two-dimensional (2D) numerical data obtained at the Rayleigh number Ra=10^8 and the Prandtl number Pr=4.4 of an Oberbeck-Boussinesq flow with constant material parameters. The results show that most of the time, and for both velocity and temperature, the instantaneous profiles scaled by the dynamical frame method [Q. Zhou and K.-Q. Xia, Phys. Rev. Lett. 104, 104301 (2010) agree well with the classical Prandtl-Blasius laminar boundary layer (BL) profiles. Therefore, when averaging in the dynamical reference frames, which fluctuate with the respective instantaneous kinematic and thermal BL thicknesses, the obtained mean velocity and temperature profiles are also of Prandtl-Blasius type for nearly all horizontal positions. We further show that in certain situations the traditional definitions based on the time-averaged profiles can lead to unphysical BL thicknesses, while the dynamical method also in such cases can provide a well-defined BL thickness for both the kinematic and the thermal BLs.

physics.flu-dyn↗

arXiv2011

Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution

Results on the Prandtl-Blasius type kinetic and thermal boundary layer thicknesses in turbulent Rayleigh-Bénard convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl-Blasius boundary layer equations, we calculate the ratio of the thermal and kinetic boundary layer thicknesses, which depends on the Prandtl number Pr only. It is approximated as $0.588Pr^{-1/2}$ for $Pr\ll Pr^*$ and as $0.982 Pr^{-1/3}$ for $Pr^*\ll\Pr$, with $Pr^*= 0.046$. Comparison of the Prandtl--Blasius velocity boundary layer thickness with that evaluated in the direct numerical simulations by Stevens, Verzicco, and Lohse (J. Fluid Mech. 643, 495 (2010)) gives very good agreement. Based on the Prandtl--Blasius type considerations, we derive a lower-bound estimate for the minimum number of the computational mesh nodes, required to conduct accurate numerical simulations of moderately high (boundary layer dominated) turbulent Rayleigh-Bénard convection, in the thermal and kinetic boundary layers close to bottom and top plates. It is shown that the number of required nodes within each boundary layer depends on Nu and Pr and grows with the Rayleigh number Ra not slower than $\sim\Ra^{0.15}$. This estimate agrees excellently with empirical results, which were based on the convergence of the Nusselt number in numerical simulations.

physics.flu-dyn↗

arXiv2011

Flow reversals in thermally driven turbulence

We analyze the reversals of the large scale flow in Rayleigh-Bénard convection both through particle image velocimetry flow visualization and direct numerical simulations (DNS) of the underlying Boussinesq equations in a (quasi) two-dimensional, rectangular geometry of aspect ratio 1. For medium Prandtl number there is a diagonal large scale convection roll and two smaller secondary rolls in the two remaining corners diagonally opposing each other. These corner flow rolls play a crucial role for the large scale wind reversal: They grow in kinetic energy and thus also in size thanks to plume detachments from the boundary layers up to the time that they take over the main, large scale diagonal flow, thus leading to reversal. Based on this mechanism we identify a typical time scale for the reversals. We map out the Rayleigh number vs Prandtl number phase space and find that the occurrence of reversals very sensitively depends on these parameters.

physics.flu-dyn↗

arXiv2010

Prandtl-Blasius temperature and velocity boundary layer profiles in turbulent Rayleigh-Bénard convection

The shape of velocity and temperature profiles near the horizontal conducting plates in turbulent Rayleigh-Bénard convection are studied numerically and experimentally over the Rayleigh number range $10^8\lesssim Ra\lesssim3\times10^{11}$ and the Prandtl number range $0.7\lesssim Pr\lesssim5.4$. The results show that both the temperature and velocity profiles well agree with the classical Prandtl-Blasius laminar boundary-layer profiles, if they are re-sampled in the respective dynamical reference frames that fluctuate with the instantaneous thermal and velocity boundary-layer thicknesses.

physics.flu-dyn↗

arXiv2008

Flow organization in non-Oberbeck-Boussinesq Rayleigh-Benard convection in water

Non-Oberbeck-Boussinesq (NOB) effects on the flow organization in two-dimensional Rayleigh-Benard turbulence are numerically analyzed. The working fluid is water. We focus on the temperature profiles, the center temperature, the Nusselt number, and on the analysis of the velocity field. Several velocity amplitudes (or Reynolds numbers) and several kinetic profiles are introduced and studied; these together describe the various features of the rather complex flow organization. The results are presented both as functions of the Rayleigh number Ra (with Ra up to 10^8) for fixed temperature difference (Delta) between top and bottom plates and as functions of Delta ("non-Oberbeck-Boussinesqness") for fixed Ra with Delta up to 60 K. All results are consistent with the available experimental NOB data for the center temperature Tc and the Nusselt number ratio Nu_{NOB}/Nu_{OB} (the label OB meaning that the Oberbeck-Boussinesq conditions are valid). Beyond Ra ~ 10^6 the flow consists of a large diagonal center convection roll and two smaller rolls in the upper and lower corners. In the NOB case the center convection roll is still characterized by only one velocity scale.

physics.flu-dyn↗

arXiv2008

Large scale dynamics in turbulent Rayleigh-Benard convection

The progress in our understanding of several aspects of turbulent Rayleigh-Benard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large-scale convection-roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Benard system.

physics.flu-dyn↗

arXiv2008

Non-Oberbeck-Boussinesq effects in turbulent thermal convection in ethane close to the critical point

As shown in earlier work (Ahlers et al., J. Fluid Mech. 569, p.409 (2006)), non-Oberbeck Boussinesq (NOB) corrections to the center temperature in turbulent Rayleigh-Benard convection in water and also in glycerol are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient. If the working fluid is ethane close to the critical point the origin of non-Oberbeck-Boussinesq corrections is very different, as will be shown in the present paper. Namely, the main origin of NOB corrections then lies in the strong temperature dependence of the isobaric thermal expansion coefficient β(T). More precisely, it is the nonlinear T-dependence of the density ρ(T) in the buoyancy force which causes another type of NOB effect. We demonstrate that through a combination of experimental, numerical, and theoretical work, the latter in the framework of the extended Prandtl-Blasius boundary layer theory developed in Ahlers et al., J. Fluid Mech. 569, p.409 (2006). The latter comes to its limits, if the temperature dependence of the thermal expension coefficient β(T) is significant.

physics.flu-dyn↗

arXiv2007

Non-Oberbeck-Boussinesq effects in two-dimensional Rayleigh-Benard convection in glycerol

We numerically analyze Non-Oberbeck-Boussinesq (NOB) effects in two-dimensional Rayleigh-Benard flow in glycerol, which shows a dramatic change in the viscosity with temperature. The results are presented both as functions of the Rayleigh number (Ra) up to $10^8$ (for fixed temperature difference between the top and bottom plates) and as functions of "non-Oberbeck-Boussinesqness'' or "NOBness'' ($Δ$) up to 50 K (for fixed Ra). For this large NOBness the center temperature $T_c$ is more than 5 K larger than the arithmetic mean temperature $T_m$ between top and bottom plate and only weakly depends on Ra. To physically account for the NOB deviations of the Nusselt numbers from its Oberbeck-Boussinesq values, we apply the decomposition of $Nu_{NOB}/Nu_{OB}$ into the product of two effects, namely first the change in the sum of the top and bottom thermal BL thicknesses, and second the shift of the center temperature $T_c$ as compared to $T_m$. While for water the origin of the $Nu$ deviation is totally dominated by the second effect (cf. Ahlers et al., J. Fluid Mech. 569, pp. 409 (2006)) for glycerol the first effect is dominating, in spite of the large increase of $T_c$ as compared to $T_m$.

physics.flu-dyn↗

arXiv2002

On geometry effects in Rayleigh-Benard convection

Various recent experiments hint at a geometry dependence of scaling relations in Rayleigh-Bénard convection. Aspect ratio and shape dependences have been found. In this paper a mechanism is offered which can account for such dependences. It is based on Prandtl's theory for laminar boundary layers and on the conservation of volume flux of the large scale wind. The mechanism implies the possibility of different thicknesses of the kinetic boundary layers at the sidewalls and the top/bottom plates, just as experimentally found by Qiu and Xia (Phys. Rev. E58, 486 (1998)), and also different $Ra$-scaling of the wind measured over the plates and at the sidewalls. In the second part of the paper a scaling argument for the velocity and temperature fluctuations in the bulk is developeVarious recent experiments hint at a geometry dependence of scaling relations in Rayleigh-Benard convection. Aspect ratio and shape dependences have been found. In this paper a mechanism is offered which can account for such dependences. It is based on Prandtl's theory for laminar boundary layers and on the conservation of volume flux of the large scale wind. The mechanism implies the possibility of different thicknesses of the kinetic boundary layers at the sidewalls and the top/bottom plates, just as experimentally found by Qiu and Xia (Phys. Rev. E58, 486 (1998)), and also different $Ra$-scaling of the wind measured over the plates and at the sidewalls. In the second part of the paper a scaling argument for the velocity and temperature fluctuations in the bulk is develope

nlin.CD↗

arXiv2000

Thermal convection for large Prandtl numbers

The Rayleigh-Benard theory by Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)] is extended towards very large Prandtl numbers Pr. The Nusselt number Nu is found here to be independent of Pr. However, for fixed Rayleigh numbers Ra> 10^{10} a maximum around Pr= 2 in the Nu(Pr)-dependence is predicted which is absent for lower Ra. We moreover offer the full functional dependences of Nu(Ra,Pr) and Re(Ra,Pr) within this extended theory, rather than only giving the limiting power laws as done previously. This enables us to more realistically describe the transitions between the various scaling regimes, including their widths.

nlin.CD↗

arXiv2000

Scaling of global momentum transport in Taylor-Couette and pipe flow

We interpret measurements of the Reynolds number dependence of the torque in Taylor-Couette flow by Lewis and Swinney [Phys. Rev. E 59, 5457 (1999)] and of the pressure drop in pipe flow by Smits and Zagarola, [Phys. Fluids 10, 1045 (1998)] within the scaling theory of Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)], developed in the context of thermal convection. The main idea is to split the energy dissipation into contributions from a boundary layer and the turbulent bulk. This ansatz can account for the observed scaling in both cases if it is assumed that the internal wind velocity $U_w$ introduced through the rotational or pressure forcing is related to the the external (imposed) velocity U, by $U_w/U \propto Re^ξ$ with xi = -0.051 and xi = -0.041 for the Taylor-Couette (U inner cylinder velocity) and pipe flow (U mean flow velocity) case, respectively. In contrast to the Rayleigh-Benard case the scaling exponents cannot (yet) be derived from the dynamical equations.

nlin.CD↗

arXiv1999

Scaling in thermal convection: A unifying theory

A systematic theory for the scaling of the Nusselt number $Nu$ and of the Reynolds number $Re$ in strong Rayleigh-Benard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large scale convection roll (``wind of turbulence'') and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number versus Prandtl number phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra smaller than 10^{11}) the leading terms are $Nu\sim Ra^{1/4}Pr^{1/8}$, $Re \sim Ra^{1/2} Pr^{-3/4}$ for $Pr < 1$ and $Nu\sim Ra^{1/4}Pr^{-1/12}$, $Re \sim Ra^{1/2} Pr^{-5/6}$ for $Pr > 1$. In most measurements these laws are modified by additive corrections from the neighboring regimes so that the impression of a slightly larger (effective) Nu vs Ra scaling exponent can arise. -- The presented theory is best summarized in the phase diagram figure 1.

chao-dyn↗

arXiv1998

Continued Fraction Representation of Temporal Multi Scaling in Turbulence

It was shown recently that the anomalous scaling of simultaneous correlation functions in turbulence is intimately related to the breaking of temporal scale invariance, which is equivalent to the appearance of infinitely many times scales in the time dependence of time-correlation functions. In this paper we derive a continued fraction representation of turbulent time correlation functions which is exact and in which the multiplicity of time scales is explicit. We demonstrate that this form yields precisely the same scaling laws for time derivatives and time integrals as the "multi-fractal" representation that was used before. Truncating the continued fraction representation yields the "best" estimates of time correlation functions if the given information is limited to the scaling exponents of the simultaneous correlation functions up to a certain, finite order. It is worth noting that the derivation of a continued fraction representation obtained here for an operator which is not Hermitian or anti-Hermitian may be of independent interest.

chao-dyn↗

arXiv1998

Analysis of Rayleigh-Plesset dynamics for sonoluminescing bubbles

Recent work on single bubble sonoluminescence (SBSL) has shown that many features of this phenomenon, especially the dependence of SBSL intensity and stability on experimental parameters, can be explained within a hydrodynamic approach. More specifically, many important properties can already be derived from an analysis of bubble wall dynamics. This dynamics is conveniently described by the Rayleigh-Plesset (RP) equation. In this work we derive analytical approximations for RP dynamics and subsequent analytical laws for parameter dependences. These results include (i) an expression for the onset threshold of SL, (ii) an analytical explanation of the transition from diffusively unstable to stable equilibria for the bubble ambient radius (unstable and stable sonoluminescence), and (iii) a detailed understanding of the resonance structure of the RP equation. It is found that the threshold for SL emission is shifted to larger bubble radii and larger driving pressures if surface tension is enlarged, whereas even a considerable change in liquid viscosity leaves this threshold virtually unaltered. As an enhanced viscosity stabilizes the bubbles against surface oscillations, we conclude that the ideal liquid for violently collapsing, surface stable SL bubbles should have small surface tension and large viscosity, although too large viscosity (>40 times the viscosity of water) will again preclude collapses.

chao-dyn↗

arXiv1998

Improved variational principle for bounds on energy dissipation in turbulent shear flow

We extend the Doering-Constantin approach to upper bounds on energy dissipation in turbulent flows by introducing a balance parameter into the variational principle. This parameter governs the relative weight of different contributions to the dissipation rate. Its optimization leads to improved bounds without entailing additional technical difficulties. For plane Couette flow, the high-Re-bounds obtainable with one-dimensional background flows are methodically lowered by a factor of 27/32.

cond-mat.soft↗

arXiv1998

Variational bound on energy dissipation in plane Couette flow

We present numerical solutions to the extended Doering-Constantin variational principle for upper bounds on the energy dissipation rate in turbulent plane Couette flow. Using the compound matrix technique in order to reformulate this principle's spectral constraint, we derive a system of equations that is amenable to numerical treatment in the entire range from low to asymptotically high Reynolds numbers. Our variational bound exhibits a minimum at intermediate Reynolds numbers, and reproduces the Busse bound in the asymptotic regime. As a consequence of a bifurcation of the minimizing wavenumbers, there exist two length scales that determine the optimal upper bound: the effective width of the variational profile's boundary segments, and the extension of their flat interior part.

cond-mat.soft↗

arXiv1998

Variational bound on energy dissipation in turbulent shear flow

We present numerical solutions to the extended Doering-Constantin variational principle for upper bounds on the energy dissipation rate in plane Couette flow, bridging the entire range from low to asymptotically high Reynolds numbers. Our variational bound exhibits structure, namely a pronounced minimum at intermediate Reynolds numbers, and recovers the Busse bound in the asymptotic regime. The most notable feature is a bifurcation of the minimizing wavenumbers, giving rise to simple scaling of the optimized variational parameters, and of the upper bound, with the Reynolds number.

cond-mat.soft↗

arXiv1997

Nonperturbative renormalization group approach to turbulence

We suggest a new, renormalization group (RG) based, nonperturbative method for treating the intermittency problem of fully developed turbulence which also includes the effects of a finite boundary of the turbulent flow. The key idea is not to try to construct an elimination procedure based on some assumed statistical distribution, but to make an ansatz for possible RG transformations and to pose constraints upon those, which guarantee the invariance of the nonlinear term in the Navier-Stokes equation, the invariance of the energy dissipation, and other basic properties of the velocity field. The role of length scales is taken to be inverse to that in the theory of critical phenomena; thus possible intermittency corrections are connected with the outer length scale. Depending on the specific type of flow, we find different sets of admissible transformations with distinct scaling behavior: for the often considered infinite, isotropic, and homogeneous system K41 scaling is enforced, but for the more realistic plane Couette geometry no restrictions on intermittency exponents were obtained so far.

chao-dyn↗