arXiv subjects

Siegfried Grossmann

Publications and source records attributed to Siegfried Grossmann.

47 records · Page 3Linked to original sources

arXiv1997

Maxwell's Demon at work: Two types of Bose condensate fluctuations in power-law traps

After discussing the key idea underlying the Maxwell's Demon ensemble, we employ this idea for calculating fluctuations of ideal Bose gas condensates in traps with power-law single-particle energy spectra. Two essentially different cases have to be distinguished. If the heat capacity remains continuous at the condensation point in the large-N-limit, the fluctuations of the number of condensate particles vanish linearly with temperature, independent of the trap characteristics. If the heat capacity becomes discontinuous, the fluctuations vanish algebraically with temperature, with an exponent determined by the trap. Our results are based on an integral representation that yields the solution to both the canonical and the microcanonical fluctuation problem in a singularly transparent manner.

cond-mat.stat-mech↗

arXiv1997

Sound radiation of 3 MHz driven gas bubbles

The sound radiation of 3 MHz acoustically driven air bubbles in liquid is analysed with respect to possible applications in second harmonic ultrasound diagnostics devices, which have recently come into clinical use. In the forcing pressure amplitude P_a = 1-10 atm and ambient radius R_0 = 0.5-5 μm parameter domain a narrow regime around the resonance radius R_0 \sim 1-1.5 μm and relatively modest P_a \sim 2-2.5 atm is identified in which optimal sound yield in the second harmonic is achieved while maintaining spherical stability of the bubble. For smaller P_a and larger R_0 hardly any sound is radiated; for larger P_a bubbles become unstable towards non-spherical shape oscillations of their surface. The computation of these instabilities is essential for the evaluation of the optimal parameter regime. A region of slightly smaller R_0 and P_a \sim 1-3 atm is best suited to achieve large ratios of the second harmonic to the fundamental intensity. Spherical stability is guaranteed in the suggested regimes for liquids with an enhanced viscosity compared to water, such as blood.

chao-dyn↗

arXiv1997

From number theory to statistical mechanics: Bose-Einstein condensation in isolated traps

We question the validity of the grand canonical ensemble for the description of Bose-Einstein condensation of small ideal Bose gas samples in isolated harmonic traps. While the ground state fraction and the specific heat capacity can be well approximated with the help of the conventional grand canonical arguments, the calculation of the fluctuation of the number of particles contained in the condensate requires a microcanonical approach. Resorting to the theory of restricted partitions of integer numbers, we present analytical and numerical results for such fluctuations in one- and three-dimensional traps, and show that their magnitude is essentially independent of the total particle number.

cond-mat.stat-mech↗

arXiv1997

Different intermittency for longitudinal and transversal turbulent fluctuations

Scaling exponents of the longitudinal and transversal velocity structure functions in numerical Navier-Stokes turbulence simulations with Taylor-Reynolds numbers up to $\rel = 110$ are determined by the extended self similarity method. We find significant differences in the degree of intermittency: For the sixth moments the scaling corrections to the classical Kolmogorov expectations are $δξ_6^L= -0.21 \pm 0.01$ and $\dx_6^T= -0.43 \pm 0.01$, respectively, independent of $\rel$. Also the generalized extended self similarity exponents $ρ_{p,q} = \dx_p/\dx_q$ differ significantly for the longitudinal and transversal structure functions. Within the She-Leveque model this means that longitudinal and transversal fluctuations obey different types of hierarchies of the moments. Moreover, the She-Leveque model hierarchy parameters $β^L $ and $β^T$ show small but significant dependences on the order of the moment.

chao-dyn↗

arXiv1997

Scaling of the irreducible SO(3)-invariants of velocity correlations in turbulence

The scaling behavior of the SO(3) irreducible amplitudes $d_n^l(r)$ of velocity structure tensors (see L'vov, Podivilov, and Procaccia, Phys. Rev. Lett. (1997)) is numerically examined for Navier-Stokes turbulence. Here, l characterizes the irreducible representation by the index of the corresponding Legendre polynomial, and n denotes the tensorial rank, i.e., the order of the moment. For moments of different order n but with the same representation index l extended self similarity (ESS) towards large scales is found. Intermittency seems to increase with l. We estimate that a crossover behavior between different inertial subrange scaling regimes in the longitudinal and transversal structure functions will hardly be detectable for achievable Reynolds numbers.

chao-dyn↗

arXiv1997

Application of extended self similarity in turbulence

From Navier-Stokes turbulence numerical simulations we show that for the extended self similarity (ESS) method it is essential to take the third order structure function taken with the modulus and called $D_3^*(r)$, rather than the standard third order structure function $D_3(r)$ itself. If done so, we find ESS towards scales larger than roughly 10 eta, where eta is the Kolmogorov scale. If $D_3(r)$ is used, there is no ESS. We also analyze ESS within the Batchelor parametrization of the second and third order longitudinal structure function and focus on the scaling of the transversal structure function. The Re-asymptotic inertial range scaling develops only beyond a Taylor-Reynolds number of about 500.

chao-dyn↗

arXiv1996

Developed turbulence: From full simulations to full mode reductions

Developed Navier-Stokes turbulence is simulated with varying wavevector mode reductions. The flatness and the skewness of the velocity derivative depend on the degree of mode reduction. They show a crossover towards the value of the full numerical simulation when the viscous subrange starts to be resolved. The intermittency corrections of the scaling exponents of the pth order velocity structure functions seem to depend mainly on the proper resolution of the inertial subrange. Universal scaling properties (i.e., independent of the degree of mode reduction) are found for the relative scaling exponents rho which were recently defined by Benzi et al.

chao-dyn↗

arXiv1994

Finite size corrections to scaling in high Reynolds number turbulence

We study analytically and numerically the corrections to scaling in turbulence which arise due to the finite ratio of the outer scale $L$ of turbulence to the viscous scale $η$, i.e., they are due to finite size effects as anisotropic forcing or boundary conditions at large scales. We find that the deviations $\dzm$ from the classical Kolmogorov scaling $ζ_m = m/3$ of the velocity moments $\langle |\u(\k)|^m\rangle \propto k^{-ζ_m}$ decrease like $δζ_m (Re) =c_m Re^{-3/10}$. Our numerics employ a reduced wave vector set approximation for which the small scale structures are not fully resolved. Within this approximation we do not find $Re$ independent anomalous scaling within the inertial subrange. If anomalous scaling in the inertial subrange can be verified in the large $Re$ limit, this supports the suggestion that small scale structures should be responsible, originating from viscosity either in the bulk (vortex tubes or sheets) or from the boundary layers (plumes or swirls).

chao-dyn↗

arXiv1993

Universality in fully developed turbulence

We extend the numerical simulations of She et al. [Phys.\ Rev.\ Lett.\ 70, 3251 (1993)] of highly turbulent flow with $15 \le$ Taylor-Reynolds number $Re_λ\le 200$ up to $Re_λ\approx 45000$, employing a reduced wave vector set method (introduced earlier) to approximately solve the Navier-Stokes equation. First, also for these extremely high Reynolds numbers $Re_λ$, the energy spectra as well as the higher moments -- when scaled by the spectral intensity at the wave number $k_p$ of peak dissipation -- can be described by {\it one universal} function of $k/k_p$ for all $Re_λ$. Second, the ISR scaling exponents $ζ_m$ of this universal function are in agreement with the 1941 Kolmogorov theory (the better, the large $Re_λ$ is), as is the $Re_λ$ dependence of $k_p$. Only around $k_p$ viscous damping leads to slight energy pileup in the spectra, as in the experimental data (bottleneck phenomenon).

chao-dyn↗

arXiv1993

Scale resolved intermittency in turbulence

The deviations $δζ_m$ ("intermittency corrections") from classical ("K41") scaling $ζ_m=m/3$ of the $m^{th}$ moments of the velocity differences in high Reynolds number turbulence are calculated, extending a method to approximately solve the Navier-Stokes equation described earlier. We suggest to introduce the notion of scale resolved intermittency corrections $δζ_m(p)$, because we find that these $δζ_m(p)$ are large in the viscous subrange, moderate in the nonuniversal stirring subrange but, surprisingly, extremely small if not zero in the inertial subrange. If ISR intermittency corrections persisted in experiment up to the large Reynolds number limit, our calculation would show, that this could be due to the opening of phase space for larger wave vectors. In the higher order velocity moment $\langle|u(p)|^m\rangle$ the crossover between inertial and viscous subrange is $(10ηm/2)^{-1}$, thus the inertial subrange is {\it smaller} for higher moments.

chao-dyn↗

arXiv1993

Fractal dimension crossovers in turbulent passive scalar signals

The fractal dimension $δ_g^{(1)}$ of turbulent passive scalar signals is calculated from the fluid dynamical equation. $δ_g^{(1)}$ depends on the scale. For small Prandtl (or Schmidt) number $Pr<10^{-2}$ one gets two ranges, $δ_g^{(1)}=1$ for small scale r and $δ_g^{(1)}$=5/3 for large r, both as expected. But for large $Pr> 1$ one gets a third, intermediate range in which the signal is extremely wrinkled and has $δ_g^{(1)}=2$. In that range the passive scalar structure function $D_θ(r)$ has a plateau. We calculate the $Pr$-dependence of the crossovers. Comparison with a numerical reduced wave vector set calculation gives good agreement with our predictions.

chao-dyn↗