G. Grötzbach

Spatial resolution requirements for direct numerical simulation of the Rayleigh-Bénard convection

Abstract Three criteria are deduced for the prediction of grids, which allow for accurate direct numerical simulations of turbulent flows. These criteria are based on wavelength considerations, boundary layer thickness estimates, and on a simplified theoretical model to calculate the coefficient of a verified subgrid-scale heat flux model. The criteria have been successfully tested by comparing the results of several three-dimensional and time-dependent numerical simulations for the Rayleigh-Benard convection of air in an infinite channel up to Ra = 381,225. Numerical results deduced from appropriate grids are in agreement with adequate experimental data. Numerical results deduced from insufficient grids show only weak deficiencies. The most sensitive data to restricted large wavelengths are the calculated Nusselt numbers and the flow regimes in the laminar-turbulent transition range; data sensitive to insufficient vertical resolution near the walls are also the Nusselt numbers; and data sensitive to insufficient horizontal resolution are the calculated Nusselt numbers and rms values of velocity and temperature fluctuations at large Rayleigh numbers. All three criteria use data specific to the type of flow only as input parameters. Therefore, these criteria may also be used for other types of flows.

Direct numerical and large eddy simulations in nuclear applications

Direct numerical and large eddy simulations are powerful tools for analyses of turbulent flows at low and large Reynolds or Rayleigh numbers in fundamental research. The current status of both methods and recent extensions are compiled. The progress achieved with subgrid scale models and numerics makes the method attractive for investigations in nuclear research and engineering. Applications of both methods of realistic technical flows are discussed. Open problems are mainly related to more general subgrid scale models for large complex containers, to the wall and inlet conditions for high Reynolds number and buoyant flows, and to discretisation schemes for local refinement of spatial resolution. As a classical example for the use of direct simulations, results are presented for a turbulent internally heated horizontal fluid layer. The analysis of the closure terms in the transport equation of the kinetic energy demonstrates major difficulties of the conventional statistical modelling for partially stably stratified convection.

Numerical simulation of turbulent temperature fluctuations in liquid metals

Abstract The method of direct numerical simulation is used to investigate temperature fluctuations in fully developed turbulent liquid metal flows. Subgrid scale models using one transport equation account for the turbulence not resolved by the finite difference grid. A special subgrid scale heat flux model for liquid metal flows is deduced together with a method of calculating the model coefficients. At very small Peclet numbers the temperatures become independent of model parameters. Numerical results for the Nusselt number in plane channels and for radial temperature and eddy conductivity profiles in annuli agree with published data. Nusselt numbers determined numerically for annuli indicate that many empirical correlations overestimate the influence of the ratio of radii. The numerical results for the eddy conductivity profiles may be used to reduce these problems. The statistical properties of the temperature fluctuations simulated are within the scatter band of experimental data. The numerical results confirm Lawns theory, giving reasonable heat flux correlation coefficients which depend only weakly on the problem marking parameters.

Analysis and modelling of the temperature variance equation in turbulent natural convection for low-Prandtl-number fluids

Results of direct numerical simulation (DNS) for Rayleigh-Benard convection for the Prandtl number Pr=0.025 are used to show some peculiarities of turbulent natural convection for low-Prandtl-number fluids. Simulations for this flow at sufficiently large Rayleigh numbers became feasible only recently because this flow requires the resolution of very small velocity scales and the recording of long-wave structures for the slow changes in the convective temperature field

Direct Numerical Simulation of Turbulent Velocity, Pressure, and Temperature Fields in Channel Flows

A finite difference scheme for direct numerical simulation of turbulent velocity, pressure, and temperature fields in plane channels and annuli is described. The fluid is incompressible and has constant density and diffusivities. The method is an extended and revised version of an earlier one. It now includes simultaneous simulation of the temperature field and employs a revised subgrid scale model which has been extended to allow for moderately high Reynolds numbers (Re > 10,000) and poorly resolving grids. The purpose of this paper is to report and demonstrate the improved capabilities of the method.

Direct numerical simulation of turbulence in an internally heated convective fluid layer and implications for statistical modelling

Direct numerical simulations (DNS) are reported for the convection in an internally heated convective fluid layer which is bounded by rigid isothermal horizontal walls at equal temperature. The simulations for a fluid Prandtl number of seven cover seven distinct internal Rayleigh numbers in the range 105 ≤ RaI ≤ 109. From the numerical database the changes of convective patterns and dynamics for increasing RaI , i.e. increasing turbulence intensity, are analysed. To support the development and improvement of statistical turbulence models for this special type of convection, turbulence data for mean and fluctuating temperature and velocities are provided. For the simulation with RaI = 108 budgets of turbulence kinetic energy k and vertical turbulent heat flux are presented. In addition, closure assumptions commonly used in statistical turbulence models are tested against the DNS data. It is found that the turbulent diffusive transport of k and is strongly underestimated by standard models. The modelling of...

Pressure transport in direct numerical simulations of turbulent natural convection in horizontal fluid layers

Abstract Direct numerical simulation (DNS) data of two types of turbulent natural convection in horizontal fluid layers are used to compute turbulent diffusive transport terms for turbulence kinetic energy and vertical turbulent heat flux. For both quantities turbulent diffusive transport is represented by a pressure correlation and a triple correlation. While for Rayleigh–Benard convection in air the pressure correlation dominates the triple correlation, for the convection in an internally heated layer the opposite behaviour is observed. The dominance of pressure transport in the Rayleigh–Benard convection and its minor importance in the internally heated layer is explained by the coherent structures and dynamics of the respective flow. The coherent structures are intermittent; they exist only for limited time intervals. Thus, conventional closure relations for turbulent diffusive transport, which are used in statistical turbulence models basing on long-time averaged quantities, may not be appropriate for the flows under consideration.

Consistent modelling of fluctuating temperature-gradient-velocity-gradient correlations for natural convection

A model is proposed for the buoyant production term in the dissipation rate equation of turbulence kinetic energy and the molecular sink term in the turbulent heat flux equation. Based on an analytical decomposition by a two-point correlation technique, the model consists of an inhomogeneous and a homogeneous part. The inhomogeneous part involves the Laplacian operator of the turbulent heat flux and needs no further modeling. For the homogeneous part of the model is derived that incorporates Pr/R as key parameter, where Pr is the Prandtl number and R is the ratio of thermal to mechanical turbulent time scales. The model is shown to obey the correct wall-limiting behaviour without further wall corrections. Comparisons with (DNS) data for Rayleigh Benard convection in air and sodium, and for convection in an internally heated fluid layer confirm its excellent near-wall performance for a wide range of Prandtl numbers. Utilizing these DNS data, the performance of the model in the bulk region is improved by slightly modifying the homogeneous part of the model.

Numerical investigation of radial mixing capabilities in strongly buoyancy-influenced vertical, turbulent channel flows

Abstract The flow behavior in the HDR downcomer during setting of the initial conditions for blowdown tests is investigated with the numerical simulation program for turbulent channel flows, TURBIT-3. This computer code is based on the complete 3-dimensional non-stationary basic equations for mass, momentum and heat. The subgrid scale models used for the turbulence structures not directly resolved by the grid are extended to take into account the buoyancy in the case of turbulent channel flow. The extended computer code is used to investigate how fast differences in temperature can be reduced, which are caused by inadequate mixing in the lower plenum during upward flow in the downcomer under conditions of mixed convection. It appears that, contrary to the computations neglecting the influences of buoyancy, the temperature differences are rapidly reduced already in the entrance zone of the downcomer. In this zone, local recirculation takes place in the cold region, which is quickly suppressed with increasing distance from the entrance by the intensification of the turbulence effects. A hot chimney extending through the whole downcomer cannot develop. Already at half level, the influence of buoyancy can be considered to be negligible in the downcomer which is assumed adiabatic. Under these conditions it should be possible in principle to set the enthalpy stratification by the planned layout of the experiment in the HDR-pressure vessel.

Turbulent heat flux and temperature variance dissipation rate in natural convection in lead-bismuth

Abstract Results of a direct numerical simulation (DNS) for Rayleigh-Bénard convection for the Rayleigh number Ra = 105 in a fluid with the Prandtl number Pr = 0.025, which corresponds to liquid lead-bismuth, are used to analyze the turbulent heat flux and the temperature variance dissipation rate. The results indicate that application of a thermal or a mixed timescale may considerably improve gradient diffusion and algebraic heat flux models at these Rayleigh and Prandtl numbers. Therefore, a good approximation of the temperature variance dissipation rate is required. The standard temperature variance dissipation rate model is investigated using the DNS results. The analysis of the standard model shows the importance of wall functions and qualitatively good predictions by the model for this type of flow. Quantitatively, the model overpredicts the temperature variance dissipation rate evaluated from the results of DNS by ˜25%. The two-point correlation method is used to derive new models for the temperature variance dissipation rate. Comparison with DNS results shows qualitatively and quantitatively good predictions by the new models. These new models lead therefore to an increased accuracy of the turbulent heat flux models for this type of flow.