TY - GEN A1 - Helling, Christiane A1 - Klein, Rupert A1 - Woitke, Peter A1 - Sedlmayr, Erwin T1 - Dust formation in brown dwarf atmospheres under conditions of driven turbulence N2 - Based on the knowledge gained from direct numerical simulations which are only possible in the microscale regime, a concept of driven turbulence is presented which allows to enter the mesoscopic scale regime. Here, dust formation under stochastic hydro- and thermodynamic conditions is studied: constructively superimposed stochastic waves initiate dust formation by the creation of singular nucleation events. It, hence, results a varying mean grain size and dust density in space and time. The newly formed dust changes the thermodynamic behavior from almost isotherm to adiabatic and chemically depletes the gas phase. T3 - ZIB-Report - 03-07 KW - dust formation KW - turbulence modelling KW - stochastic boundaries Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7294 ER - TY - GEN A1 - Ruprecht, Daniel A1 - Klein, Rupert A1 - Majda, Andrew J. T1 - Moisture-Gravity Wave Interactions in a Multiscale Environment N2 - Starting from the conservation laws for mass, momentum and energy together with a three species, bulk microphysic model, a model for the interaction of internal gravity waves and deep convective hot towers is derived by using multiscale asymptotic techniques. From the resulting leading order equations, a closed model is obtained by applying weighted averages to the smallscale hot towers without requiring further closure approximations. The resulting model is an extension of the linear, anelastic equations, into which moisture enters as the area fraction of saturated regions on the microscale with two way coupling between the large and small scale. Moisture reduces the effective stability in the model and defines a potential temperature sourceterm related to the net effect of latent heat release or consumption by microscale up- and downdrafts. The dispersion relation and group velocity of the system is analyzed and moisture is found to have several effects: It reduces energy transport by waves, increases the vertical wavenumber but decreases the slope at which wave packets travel and it introduces a lower horizontal cutoff wavenumber, below which modes turn into evanescent. Further, moisture can cause critical layers. Numerical examples for steadystate and timedependent mountain waves are shown and the effects of moisture on these waves are investigated. T3 - ZIB-Report - 09-21 Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11358 SN - 1438-0064 ER - TY - GEN A1 - Carqué, Gunter A1 - Owinoh, Antony Z. A1 - Klein, Rupert A1 - Majda, Andrew J. T1 - Asymptotic Scale Analysis of Precipitating Clouds N2 - Asymptotic analyses of the three dimensional compressible flow equations coupled with transport equations for the mixing ratios of water vapour, cloud water and rain water are described. We obtain reduced systems of equations for two particular regimes of length and time scales: Models for the long time evolution of deep convective columns and for the short time evolution of shallow convective layers. The asymptotic deep convective column model is anelastic, yet the vertical motion is pressure free, i.e., it evolves freely in interaction with buoyancy while the horizontal divergence adjusts to fullfil the anelastic constraint. The perturbation pressure guaranteeing compliance with the horizontal divergence constraint obeys a Poisson-type equation. Surprisingly, the vertical velocity plays an important role in the horizontal dynamics through the Coriolis term. The vertical acceleration in a saturated column is directly determined by the buoyancy induced by potential temperature differences relative to the background stratification. This potential temperature deviation is a conserved quantity. Evaporation is the only important microphysical process in the undersaturated regime. The evaporation rate depends on the saturation deficit and the amount of rain water present and determines the (downward) vertical velocity and the distribution of water vapour. To connect the deep convective column solutions to top and bottom boundary conditions, a different flow regime needs to be accounted for. Within shallow layers whose depth is comparable to the column diameters, adjustment to physical boundary conditions can take place. This is the second regime considered in this report. The shallow convective layer regime is shown to be asymptotically described by Boussinesq-type equations. These equations are closed by evolution equations which show that, in the saturated regime, the distributions of potential temperature and cloud water are determined by a condensation rate that is directly proportional to the vertical velocity. In the undersaturated regime, the potential temperature distribution is determined by the amount of rain present, since the water vapour in this case is shown to be a conserved quantity. In both regimes the distribution of rain water depends on the rain water flux. T3 - ZIB-Report - 08-03 KW - atmospheric moist convection KW - deep convection KW - shallow convection KW - precipitating clouds KW - asymptotic analysis Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10548 SN - 1438-0064 ER - TY - GEN A1 - Carqué, Gunter A1 - Schmidt, Heiko A1 - Stevens, Bjorn A1 - Klein, Rupert T1 - Plausibility Check of an Asymptotic Column Model for Deep Convective Clouds N2 - By use of asymptotic analysis Carqué et al. [ZIB-Report 08-03] derived an asymptotic column model for deep convective clouds based on the three dimensional compressible flow equations and a bulk microphysics parameterization. In the present study we check the plausibility of the reduced model equations by comparing implications of the model for the scaling of various terms in the governing equations with those extracted from large eddy simulation data based on the computational model UCLA-LES1.1. This code solves an anelastic system of equations with complete droplet based microphysics and LES closures. We observe that the simulation data corroborate the basic assumptions of the asymptotic analysis and the main conclusions implied by the asymptotically reduced model. The code output reflects the scales of space and time: The deep convective clouds show an anisotropic structure where the horizontal scale is considerably narrower than the vertical scale; with a period of about 20 min, from emergence to breakup, the life cycle of one particular deep convective cloud corresponds exactly to the reference time of the reduced model. The characteristic properties of dynamics as predicted by the reduced model are also reflected in the simulation data: The horizontal flow is controlled by the pressure field; the vertical velocity develops freely independent of pressure over the depth of the convective column; the vertical velocity is directly determined by the buoyancy induced by the potential temperature deviation relative to the background stratification. With respect to grid resolution we observe that refining the spatial step size of the equidistant computational grid from 125 m to 62.5 m does not influence the results: Even with the coarser grid the relevant physical phenomena are sufficiently resolved. Somewhat surprisingly, the Coriolis term involving vertical velocity and acting on the horizontal (east-west) velocity component appears at leading order in the asymptotics. Accordingly, we expected to find a nontrivial impact of this Coriolis effect on the horizontal flow velocity components within columns of updrafts. However, switching the term on and off in subsequent simulations did not sizeably affect the results. T3 - ZIB-Report - 08-44 KW - atmospheric moist convection KW - deep convection KW - precipitating clouds KW - asymptotic analysis KW - Large Eddy Simulation Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10967 SN - 1438-0064 ER - TY - GEN A1 - Oevermann, Michael A1 - Scharfenberg, Carsten A1 - Klein, Rupert T1 - A sharp interface finite volume method for elliptic equations on Cartesian grids N2 - We present a second order sharp interface finite volume method for the solution of the three-dimensional poisson equation with variable coefficients on Cartesian grids. In particular, we focus on interface problems with discontinuities in the coefficient, the source term, the solution, and the fluxes across the interface. The method uses standard piecewiese trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27-point stencil. Singularities associated with vanishing partial volumes of intersected grid cells are removed by a two-term asymptotic approach. In contrast to the 2D method presented by two of the authors in [M.~Oevermann, R.~Klein: A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J.~Comp.~Phys.~219 (2006)] we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz. This simplifies the treatment of the different cut-cell types and avoids additional special operations for degenerated interface topologies. The resulting set of linear equations has been solved with a BiCGSTAB solver preconditioned with an algebraic multigrid. In various testcases -- including large coefficient ratios and non-smooth interfaces -- the method achieves second order of accuracy in the L_inf and L_2 norm. T3 - ZIB-Report - 08-38 KW - elliptische Gleichungen KW - Finite Volumen KW - eingebettete Ränder KW - variable und diskontinuierliche Koeffizienten KW - diskontinuierliche Lösung KW - Elliptic equations KW - finite volume methods KW - embedded interface KW - variable and discontinuous coefficients KW - discontinuous solution Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10900 SN - 1438-0064 ER - TY - GEN A1 - Vater, Stefan A1 - Klein, Rupert T1 - Stability of a Cartesian Grid Projection Method for Zero Froude Number Shallow Water Flows N2 - In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov-Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown incompressible height and piecewise constant test functions. The stability of the projection is proved using the theory of generalized mixed finite elements, which goes back to Nicola{\"i}des (1982). In order to do so, the validity of three different inf-sup conditions has to be shown. Since the zero Froude number shallow water equations have the same mathematical structure as the incompressible Euler equations of isentropic gas dynamics, the method can be easily transfered to the computation of incompressible variable density flow problems. T3 - ZIB-Report - 07-13 KW - incompressible flows KW - shallow water equations KW - projection method KW - stability KW - mixed finite elements KW - inf-sup-condition Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9562 ER - TY - GEN A1 - Oevermann, Michael A1 - Klein, Rupert T1 - A cartesian grid finite volume method for the solution of the Poisson equation with variable coefficients and embedded interfaces N2 - We present a finite volume method for the solution of the two-dimensional Poisson equation $ \nabla\cdot( \beta( {\mbox{\boldmath $x$}}) \nabla u({\mbox{\boldmath $x$}})) = f(\mbox{\boldmath $x$}) $ with variable, discontinuous coefficients and solution discontinuities on irregular domains. The method uses bilinear ansatz functions on Cartesian grids for the solution $u({\mbox{\boldmath $x$})$ resulting in a compact nine-point stencil. The resulting linear problem has been solved with a standard multigrid solver. Singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two-step asymptotic approach. The method achieves second order of accuracy in the $L^\infty$ and $L^2$ norm. T3 - ZIB-Report - 06-05 KW - Poisson equation KW - finite volume methods KW - embedded interface KW - ariable and discontinuous coefficients KW - discontinuous solution Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8996 ER - TY - GEN A1 - Schneider, Thomas A1 - Botta, Nicola A1 - Geratz, Karl Josef A1 - Klein, Rupert T1 - Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flow N2 - When attempting to compute unsteady, variable density flows at very small or zero Mach number using a standard finite volume compressible flow solver one faces at least the following difficulties: (i) Spatial pressure variations vanish as the Mach number $M \rightarrow 0$, but they do affect the velocity field at leading order; (ii) the resulting spatial homogeneity of the leading order pressure implies an elliptic divergence constraint for the energy flux; (iii) violation of this constraint would crucially affect the transport of mass, thereby disabling a code to properly advect even a constant density distribution. A previous companion paper derived the above observations from a single time - multiple length scale asymptotic analysis for $M \ll 1$, applied to the conservation form of the governing equations and assuming an ideal gas with constant specific heats. The paper then restricted to weakly compressible one-dimensional flows and introduced a semi-implicit extension of a compressible flow solver, designed to handle the interaction of long wavelength acoustics with small scale, large amplitude density fluctuations. In the present paper we concentrate on the limit of zero Mach number for multi-dimensional, variable density flows. The construction of numerical fluxes for all conserved quantities involves: An explicit upwind step (1) yielding predictions for the nonlinear convective flux components. This procedure still neglects the influence of pressure gradients on the convective fluxes during the time step. Suitable corrections are applied in step (2), which guarantees compliance of the convective fluxes with the divergence constraint. This step requires the solution of a Poisson-type equation to obtain the relevant pressure gradients. Step (3), which requires the solution of a second Poisson-type equation, yields the yet unknown (non-convective) pressure contribution to the total flux of momentum. The final, cell centered velocity field exactly satisfies a discrete divergence constraint consistent with the asymptotic limit. Notice that step (1) can be done by any standard finite volume compressible flow solver and that the input to steps (2) and (3) involves solely the fluxes from step (1), but is independent on how these were obtained. Thus, we claim that our approach allows any such solver to be extended to simulate incompressible flows. Extensions to the weakly compressible regime $0 < M \ll 1$, reactive flows and more complex equations of state will be addressed in follow-up publications. T3 - ZIB-Report - SC-98-31 KW - Incompressible variable density flows KW - conservation laws Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3749 ER - TY - GEN A1 - Oevermann, Michael A1 - Klein, Rupert A1 - Berger, Marsha A1 - Goodman, Jonathan T1 - A Projection Method for Two-Phase Incompressible Flow with Surface Tension and Sharp Interface Resolution N2 - We present a fully second order projection method for the simulation of two-phase incompressible flow with surface tension. The Navier-Stokes equations are solved with a projection method on a fixed Cartesian grid. The free interface between the two fluids is tracked with a level set approach. The conditions at the interface for the pressure, the pressure gradient, and the velocity are explicitly incorporated into the scheme leading to a sharp representation of the pressure discontinuity and the interfacial force. The scheme in the presented form does not introduce additional points in the standard finite difference stencils. Computational results are compared with analytic solutions for a static round bubble, damped surface waves, and Rayleigh-Taylor instabilities. T3 - ZIB-Report - 00-17 KW - Two-phase flow KW - immersed interface method KW - incompressible flow KW - projection method KW - level set KW - multi grid Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5851 ER - TY - JOUR A1 - Ebrahimi Viand, Roya A1 - Höfling, Felix A1 - Klein, Rupert A1 - Delle Site, Luigi T1 - Theory and simulation of open systems out of equilibrium JF - The Journal of Chemical Physics N2 - We consider the theoretical model of Bergmann and Lebowitz for open systems out of equilibrium and translate its principles in the adaptive resolution simulation molecular dynamics technique. We simulate Lennard-Jones fluids with open boundaries in a thermal gradient and find excellent agreement of the stationary responses with the results obtained from the simulation of a larger locally forced closed system. The encouraging results pave the way for a computational treatment of open systems far from equilibrium framed in a well-established theoretical model that avoids possible numerical artifacts and physical misinterpretations. Y1 - 2020 U6 - https://doi.org/10.1063/5.0014065 VL - 153 SP - 101102 ER -