A VERIFICATION AND VALIDATION APPROACH FOR EULERIAN MODELLING OF SLOSHING FLOWS

Transcription

1 A VERIFICATION AND VALIDATION APPROACH FOR EULERIAN MODELLING OF SLOSHING FLOWS C. Dinescu (1), K. Claramunt (1), B. Leonard (1), M. Mezine (1), Ch. Hirsch (1) (1) NUMECA International S.A., web page Chaussée de la Hulpe 189, B-1170 Brussels, Belgium cristian.dinescu@numeca.be ABSTRACT The paper presents the verification and validation (V&V) effort of a software tool [1] for the numerical simulation of various aspects related to the sloshing of cryogenic liquids in tanks, including the treatment of real fluids. The guidelines for structuring the V&V process are detailed followed by the description of the Eulerian Volume Of Fluid (VOF) methodology developed in the framework of the CFD flow integrated environment FINE TM /Open with OpenLabs for general complex internal/external flow simulations, with emphasis on the real fluid modelling via complex equations of state (EsOS). Computational results concerned with the thermal stratification and forced sloshing of standard and cryogenic liquids are shown. 1 Introduction The efficient management of conventional and cryogenic propellants in terms of their storage in tanks, acquisition and transfer, and their pressure and temperature control is of significant importance for operating spacecrafts/satellites according to the mission specifications. The complexity of the environmental conditions, represented by extreme imposed unsteady accelerations combined with rapid changes in the gravity level as well as various thermal effects, to which the propellant is subjected, requires mature software tools able to numerically predict the sloshing in tanks. Central to such efforts, in addition to the mathematical and numerical modelling, is the V&V of the software tools tailored for handling sloshing type phenomena associated with various phases of the spacecraft/satellite s mission. The present contribution is organized as follow: section provides details about the guidelines for the selection of the appropriate benchmarks necessary for the V&V of the main classes of physics associated with the sloshing of cryogenic propellants, while section 3 will shortly emphasize the mathematical and numerical model; sections 4 and 5 are dedicated to the computational results concerned with the thermal stratification and sloshing flows using water and cryogenic liquids. Finally, in section 6 conclusions are drawn. Verification and validation program Following the V&V paradigm of [], a similar predictive framework is to be built around a database with benchmarks currently under construction. A tier structure for the validation process derived from an analysis of the phases of a typical flight mission allowed identifying the main classes of physics to be accounted for in the verification test cases, and calibration and validation benchmarks [3]. The experimental data to be used for comparisons with the numerical simulations is provided by validation experiments (see [], [3]) characterized for the present approach by loose couplings among the classes of physics accurate description of the geometrical features of the two-phase system and the imposed initial/boundary conditions to be accounted for via the input parameters of the software tool accurate data about the properties of the two fluids estimation of the uncertainty of experimental measurements The goal of the ongoing V&V efforts is to finalize the development of dedicated modules handling accurately identified physical aspects of thermal stratification,

2 linear and non-linear sloshing of cryogenic/standard liquids with/without consideration of thermal and mass transfer effects. In what follows, the detailed computational results for benchmarks purposely designed for the validation of the thermal stratification and verification of forced sloshing flows will display the capability of the flow solver to predict the behaviours of the moving free surfaces when real fluid modelling is considered. 3 Mathematical and numerical model The mathematical model employs a single-field representation of two-phase immiscible flows. The actual governing system, employing a density based approach, is written as follows: 1 p r r + ( v) = 0 ρβ m τ (1) r r v v r r v r r r r ρ + ρ + ρ ( v ) + p τ = ρg + ρfe τ t () ρc ρc r r + + v ( ρc) = 0 τ t H r r r ρ ( H v β m )( ρ v) + τ r r r r r r r r r ρ ( vh ) = q + ( τ v) + ρ( g + f ) v where ρ, v r, p, τ, H and q r are the density, the velocity vector, the pressure, the shear stress tensor, the total enthalpy and the heat flux vector respectively. The quantities g r and f r e are the gravitational acceleration and the time varying forces respectively. The pseudo-sound speed is denoted by β. The averaged density and the m dynamic viscosity are defined in terms of VOF fraction C : e (3) (4) derivatives t are discretized with a first order backward difference formula, while for the space derivatives in the pressure advection, momentum and energy equations a central scheme stabilized with a Jameson type artificial dissipation is selected. For the VOF type equation a high compressive scheme has purposely been devised. In order to predict the pressure field, in the presence of very large density variations over a moving interface, a dedicated two-phase preconditioning technique has been constructed. For further details on the high resolution scheme and the preconditioning methodology, the formulation of particularly, we refer the reader to [4] and [1]. β m In order to fulfil the requirement of accurate representation of the sloshing fluids, mainly for the cryogenic ones, the real fluid modelling in the present software tool is managed by means of pre-computed thermodynamic tables for achieving computational efficiency. These are constructed by the dedicated TABGEN module, built on top of the REFPROP software of NIST [5], the latter one generating thermodynamic data by means of state-of-the art EsOS. During the numerical computation, the dependent quantities are looked-up in the thermos-table in function of the independent ones. In particular, the density and temperature are computed from enthalpy-pressure tables. 4 Thermal stratification This section details the validation of the numerical approach for the thermal stratification in water and emphasizes the predictive capability of the software tool for the case of liquid nitrogen LN. ( C) ρ = Cρ ρ (5) ( ) µ = Cµ C µ (6) the subscript i = 1, denoting the constituent phases of mixture. In this paper C is the VOF fraction of the dense (or liquid) phase. Concerning the time discretization a dual-time stepping strategy is used where the governing system (1) (4) is driven to steady state in the pseudo-time τ time by a 4- stage Runge-Kutta method. The physical time Figure 1. Sketch of the stratification cell

3 a) a) y/y_max=0.5 b) y/y_max=0.5 b) c) y/y_max=0.75 c) Figure. Computed temperature (a), horizontal (b) and vertical (c) components of the velocity vector For validation, a benchmark consisting of monitoring the flow evolution in a rectangular stratification cell, filled with water, has been conceived in terms of prescribing the major geometrical features of the experimental set-up, the validation objectives, the boundary conditions and the measured quantities. The main objectives were to monitor the viscous boundary layers and to evaluate the impact of the real fluid modelling via EsOS expressed as thermo-tables. Figure 3. Comparisons between computed and experimentally measured vertical velocity components at non-dimensional heights. Experimental data: courtesy of Von Karman Institute. Von Karman Institute has designed the stratification cell and executed the experimental campaign by collecting PIV experimental data, namely velocity distributions. A stratification cell with the aspect ratio (=y_max/x_max) of 1 unit has been used (see Fig. 1). The cell s vertical lateral walls have been kept at two different constant temperatures: T c and T h. The difference of temperature T=T c -T h has been varied in

4 Table 1. Reference properties of LN for thermal stratification simulations Fluid Density Temperature Pressure LN kg/m K 1e5 Pa such way that the Rayleigh number 3 max gβρ y c p T Ra = covered the domain.71e7 to µκ 3.73e7. β is the thermal expansion coefficient, specific heat capacity and κ the thermal conductivity. Fig. emphasizes the computed fields represented by the static temperatures and the horizontal (Vx) and the vertical (Vy) components of the velocity vector. The velocity fields show the presence of a counter clockwise rotating vortex responsible for the formation of the viscous and thermal boundary layers. In order to validate the numerical model, the vertical (Vy) velocity distributions recorded computationally and experimentally at the non-dimensional heights y/y_max of 0.5, 0.5 and 0.75 have been compared. The comparisons shown in Fig. 3 between the computational solutions assuming water incompressible liquid, with thermal effects handled by the Boussinesq approximation, and those using the real fluid modelling via thermo-tables prove that the software tool captures satisfactorily the viscous boundary layers. Inspecting the experimental profiles one records their asymmetry with respect to the measured negative and positive extrema. Indeed, the standard deviation σ Vy of the measured Vy-component varies along the middle horizontal axis inside boundary layers, being about 7% of the local Vy-extremum for the left peak and circa 4.4% for the right peak. This is less pronounced in the case of computed ones (see Tab.). Table. Asymmetry of the experimental and computational distribution from Fig.3 Position Experimental Computational y/y_max max max max max V yleft V yright V yleft V yright [-] V max yleft V max yleft 0.5 0% 13.7% %.3% 0/ % 4.3% c p The differences could be attributed to the modality of imposing the initial and boundary conditions in the experiments and computations. For the former, an residual unsteadiness expressed, for example, by the experimental standard deviation σ = 0.57% T at the cold side together with a geometrical uncertainty represented by a lack of parallelism of the vertical walls, namely a value of -/+1. degree communicated by the experimentalists, have not been accounted for in the shown computational results. Separate computational investigations, not shown herein, on the effect induced by the lack of parallelism confirmed that it can be neglected. In addition, a free surface has been present at the upper part of the stratification cell due to constructive constraints imposed by the necessity to control the water level. This detail has been neglected in the computations, too. The following computational results prove that in the case of LN, for reduced Ra, the usage of thermostables is of utmost importance. Two sets of computations were performed characterized by horizontally imposed temperature gradients characterized by T=0.1 o and T=9 o, corresponding to Ra=5.74e7 and Ra=5.16e9 respectively. For each set, the LN (see Tab.1) has been modelled either as incompressible liquid together with the Boussinesq approximation for considering the thermal effects or as a real fluid employing thermo-tables. Inspecting the vertical temperature distributions shown in Fig. 4(a) and Fig. 5(a), one observes no difference due to the modelling of LN. This is also reflected in the overlap of the corresponding vertical V y - distributions in Fig. 4(c) and Fig. 5(c). However, the situation is completely changed for the horizontal temperature distributions emphasized in Fig. 4(b) and Fig. 5(b). The different gradient predictions in the temperature fields, by the incompressible and real fluid modelling approaches, are automatically reflected in the prediction of the boundary layers measured by the horizontal V x -distributions in Fig. 4(d) but not in Fig. 5(d). In Fig.4(d), the extrema predicted by the real fluid modelling are lower than those in the incompressible approach. Tc c

5 a) y/y_max=0.5 a) y/y_max=0.5 b) x/x_max=0.5 b) x/x_max=0.5 c) y/y_max=0.5 c) y/y_max=0.5 d) x/x_max=0.5 Figure 4. Regime Ra=5.74e7 ( T=0.1 o ) Influence of LN modelling: incompressible liquid versus real liquid with thermo-tables: temperature distributions along the vertical (a) and horizontal (b) middle symmetry lines; V y -distribution (c) along vertical middle symmetry line and V x -distribution (d) along horizontal middle symmetry line d) x/x_max=0.5 Figure 5. Regime Ra=5.16e9 ( T=9 o ) Influence of LN modelling: incompressible liquid versus real liquid with thermo-tables: temperature distributions along the vertical (a) and horizontal (b) middle symmetry lines; V y -distribution (c) along vertical middle symmetry line and V x -distribution (d) along horizontal middle symmetry line

6 However, the situation is completely changed for the horizontal temperature distributions emphasized in Fig. 4(b) and Fig. 5(b). The different gradient predictions in the temperature fields, by the incompressible and real fluid modelling approaches, are automatically reflected in the prediction of the boundary layers measured by the horizontal V x -distributions in Fig. 4(d) but not in Fig. 5(d). In Fig.4(d), the extrema predicted by the real fluid modelling are lower than those in the incompressible approach. This difference is not found for the second regime, in Fig. 5(d), as its Ra number is two orders of magnitude higher and the convective heat transfer is dominant. Table 3. Reference properties of LN for D sloshing simulation Fluid Density Temperature Pressure LN kg/m 3 80 K e5 Pa He kg/m 3 80 K e5 Pa Table 4. Reference properties of LH for D sloshing simulation Fluid Density Temperature Pressure LH 71.5 kg/m 3 0 K e5 Pa He 4.59 kg/m 3 0 K e5 Pa 5 -D forced sloshing In order to verify the VOF methodology and the predictive capability of the software tool, a D forced sloshing flow has been investigated as in [1]. A D rectangular tank is subjected to horizontal lateral sinusoidal excitation characterized by: ( ωt) x(t ) = bsin (7) where x(t) is the abscissa of the tank, b being amplitude and ω the angular frequency of the excitation. For this configuration, Faltinsen gave in [6] the linear analytical solution in terms of the velocity potential function φ, which can be converted into the free surface displacement H( x, t) using : Figure 6. Influence of water modelling - comparison between the analytical solution and the computed free surface displacements at the vertical wall( x = -L) 1 φ H ( x, t) = g t y = 0 (8) The origin of the coordinate system is the center of the tank and it is denoted by h the water height and L the tank s length. As in [1] the water and air is considered and the influence of the real fluid modelling with thermo-tables is verified. For h = L = 0. 5m and g = 9.81m / s, the lowest natural 1 frequency is ω 0 = s. The computations are conducted in parallel, on 3 processors, on an unstructured hexahedral mesh of cells. Figure 7. Influence of real fluid modelling for LN and LH - computed free surface displacements at the vertical wall( x = -L) The mesh is refined near the interface with a grid spacing of 0.001m along vertical and horizontal axis. The imposed excitation is characterized by the following frequency and amplitude: ω = with b = m 0.99 ω0

7 In order to identify the frequency f cmp of the computed solution and compare it with f th predicted by the linear theory a fast Fourier transformation (FFT) analysis has been performed with Octave The theoretical solution shown in Fig.8 consisted of 5001 samples equally distanced in time, the time step being 0.00 sec. For the computational one, 16 samples have been employed and the time-step of 0.08 sec. The FFT analysis gave the estimated theoretical frequency f th = Hz and the computed one f cmp = Hz. This corresponds to a relative error of 0.78%. a) f th = Hz 6 3-D free sloshing In this section, the 3-D liquid sloshing in a confined tank reported in [1] is extended by employing the real fluid modelling feature. The geometrical features of the tank are its length L x, width L y. The initial shape of the free surface is given by a hump described by the following bi-dimensional displacement of the free surface: H ( x, y) = H 0 exp( β [( x Lx ) + ( y Ly ) ]), (9) b) f cmp = Hz Figure 8. FFT analysis of the theoretical (a) and computed (b) solutions for water/air fluid pairing In Fig. 6 one observes that the solution corresponding to the incompressible treatment of the water/air pair of fluids is matching well the analytical solution for the first 5 seconds and next the time period start being slightly over-predicted. As expected the real fluid modelling of the water and air as ideal gas did not introduce errors, concluding its verification. H0 is the initial height of the hump and β denotes here the peak enhancement factor. As in [1] the following settings have been employed: Lx = Ly = 10m, a water depth of h = 0. 5m, a wave amplitude H0 = 0. 05m and β = 0.4. The employed unstructured hexahedral mesh has cells with a refined box around the interface, shown in Fig.8. The space steps in these region are x = y = and z = A time step was set to t = s and simulations were performed, in parallel on 96 processors, for a time period of t = 10s. Fig. 7 presents two computational results proving the ability of the software tool to handle unsteady sloshing of LN and liquid hydrogen (LH), modelled as real fluids via the thermo-tables, when the He as ideal gas. The difference in density between LN and LH (see Tab.3 and 4) is responsible for the different amplitudes and predicted time periods. Figure 8. Zoom of the refined unstructured mesh [1]

8 8 References 1. Mezine, M., Dinescu, C., Leonard, B. & Hirsch, Ch. (011). A Density-Based Flow Solver for the Simulation of Sloshing in Tanks. Proceedings of the 7 th European Symposium on Aerothermodynamics, SP-69, August 011, ISBN ISSN X. Figure 9. Influence of real fluid modelling for HO, LN and LH - computed free surface elevation in the center of the tank Fig. 9 shows the computed solutions for the following pairs of liquids, modelled as real fluids, and gases: water/air, LN/He and LH/He. The reference properties for the latter two pairs are those used in section 5 and listed in Tab. 3 and Tab. 4. Again, the difference in density among water, LN and LH is responsible for the different amplitudes and predicted time periods. 7 Conclusion The present contribution emphasized the V&V program of a software tool dedicated to the sloshing in tanks of conventional and cryogenic propellants. The real fluid modelling of standard and cryogenic propellant has been handled via pre-computed state-ofthe art EsOS. The software tool has been validated against experimentally recorded data from a specially devised validation benchmark for the thermal stratification. The real fluid modelling of cryogenic liquids has shown its ability on predicting lower extrema for the vertical Vx-distributions compared with their modelling as incompressible fluids.. Oberkampf, W. L., Truncano, T. G., Hirsch, C. (004), Verification, Validation, and Predictive Capability in Computational Science and Physics, Applied Mechanics Review, 57 (5), Oberkampf, W. L., Truncano, T. G. (007), Verification and validation benchmarks, Nuclear Eng. Design, doi: /j.nucengdes Dinescu, C., Leonard, B. & Hirsch, Ch. (006). Eulerian Capturing Method for Two-Phase Immiscible Flows. In Book of abstracts of Euromech Colloquium 479 Numerical Simulation of Multiphase Flow with Deformable Interfaces (Eds. B. J. Boersma, E. Coyajee, L. Portella), TU Delft, The Pier, Scheveningen, The Netherlands, August NIST Reference Fluid Thermodynamic and Transport Properties Database (REFPROP), ( 6. Faltinsen, O.M. (1978). A numerical nonlinear method of sloshing in tanks with two dimensional flow. Journal Ship. Res., Finally, the predictive capability of the real fluid modelling in various D and 3D sloshing type flows has been demonstrated. The future efforts will be focused on the V&V of highly non-linear sloshing regimes, in microgravity conditions in the presence of heat and mass transfer effects.