Final Report Summary - T2T-VHF (Transition to Turbulence of Volumetrically Heated Flows)
http://www.aston.ac.uk/eas/staff/a-z/dr-sotos-generalis/(si apre in una nuova finestra)
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PUBLISABLE FINAL REPORT – Marie Sklowdoska Curie Intra European Fellowship
T2T-VHF
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This report collates the results obtained during the Marie-Curie Fellowship entitled Transition to Turbulence in Volumetrically Heated Flows (T2T-VHF - Project Reference 274367). The results are also described in more detail in references [1]-[5].
Aims
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The transition from the uniform steady laminar flow to the chaotic motion of turbulent flow is of great importance in fluid dynamics. The main motivation behind the proposed research project is to study the pre-chaotic bifurcation behaviour of strongly non-linear equilibrium solutions for incompressible volumetrically heated shear flows (VHSF) in a long channel. In particular, it is envisaged that the solutions obtained will be related to the advanced stages of transition identified by the studies involving DNS and experimental investigations in the turbulent regime.
Background
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A volumetrically heated fluid is a fluid that is heated from within. This can be heating caused by chemical or biochemical reactions or radioactive decay of elements in the earth [6] or nuclear reactors. The heat release or absorbed as a fluid component changes its physical state (i.e. from vapour to liquid or vice versa) can also be considered as volumetric heating [6]. Therefore, the phenomena we consider has a multitude of examples, which include convection in clouds and weather patterns, the transport of humidity through greenhouses (Fig. 1a, [8]) and the heating or cooling of chemical or biochemical reactors. Modelling the impact that convection has on the growth of clouds, weather patterns and severe weather, could help in their longer term prediction to enable better planning for societal responses to such weather (i.e. from disaster response and warnings to longer term flood mitigation). In the growth of clouds and greenhouses changes to the density of air occur as local changes in temperature and the water vapour content can lead to the formation of stable or unstable convection states, which can in turn affect weather formation and the growth of plants in greenhouses.
Several important industrial examples include decay heat removal in nuclear reactors, the effect that water vapour and coolant sprays have on fluid motion in the containment vessels of nuclear reactors as well as the convection of molten reactor cores that occurs in lower reactor vessel head or in core-catchers (Fig. 1b, [9]-[10]). The convection promotes mixing and it has an impact on how heat from the hot reactor or molten core is transported away from them. Understanding the impact that convection has on the turbulent flows observed during these events will help in developing mitigation strategies in order to better control undesirable situations. Molten salt nuclear reactors are also considered as volumetrically heated fluids as the fuel is dissolved in the molten salt and it can influence the turbulence observed [11].
The flow states that form at the transition between laminar convection and steady convection in volumetrically heated flows in the horizontal configuration are planforms consisting of either rolls or hexagons [1]-[5],[12]-[30]. The hexagons can either be down-welling (i.e. fluid moves down in the centre of the circulation) or up-welling (i.e. fluid moves up in the circulation centre). The states observed depend upon the nature of the fluid considered for example liquid mercury will result in the formation of up-hexagons or rolls while air and water give down-welling hexagons or rolls.
Objectives
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In the proposed study, we intend to model the transition to turbulence of VHSF. Two modelling approaches will be used to characterise natural and mixed convection. The first most accurate approach that is used by researchers in convective flows is the spectral method. The second approach utilises finite volumes to discretise the required transport equations. The key difference between the spectral code and the finite volume methods is that only the spectral code, based on the Galerkin collocation method, can resolve the eigenvalues of the problem at hand. The key advantages of the first approach are that can easily identify stable and unstable flow states and it can do so very quickly. However, the key disadvantage is that the equations used in the spectral method must be representative of the geometries considered (i.e. flat plat, rotating cylinders, spheres etc), while many engineering problems have complex geometries (i.e. chemical reactors, pipe bundles in heat exchanger, nuclear reactor cores).
Deliverables
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What we intended to do here is identify states and conditions that can be modelled accurately by both numerical methods for direct comparison. In doing this, we intend to create a toolbox that should allow us to apply appropriate amendments to the finite element code such that the relevant flow states identified by the spectral method can be also be modelled by the finite element method. More detail on the methods used and the results are reported in the corresponding subsections below. This would allow us to apply such techniques to model flows in more complex geometries that are not currently possible with spectral methods.
Methods
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Following the work plan set out in the proposal (Table 1), we used two approaches to model the convective flow states. The techniques used were spectral stability analysis and computational fluid dynamics.
• The spectral analysis involves solving the transport equations as a continuous harmonic expansion of modes (Chebyshev polynomials in the spanwise direction and Fourier modes in the longitudinal and transverse directions) that can describe the spatial dynamics of the state analysis. The states formed were controlled by the Rayleigh number, Ra, a measure of the buoyant convection in the system and the wavenumber, a non-dimensional characterisation of the size of the circulation cells which inversely proportional to the wavelength. The Prandtl number, Pr, is the dimensionless ratio of viscous to thermal diffusivity, which is 0.025 for liquid mercury, 0.705 for air and 7 for water. Therefore, Pr can control the behaviour of the fluid modelled.
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Table 1: Overview of Work Packages described on pages 6 and 7 of the proposal.
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|| Numerical Method | Work Packages ||
|| - | For Conditions of Constant Pressure Gradient and Constant Flux | Fluids (Pr) | Angle (χ) | Determine states and stable/unstable regions ||
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|| Spectral stability analysis | (i) Linear stability analysis of the basic flow | 0.01-7 | 0°-90° |Transition from laminar state to hexagons, rolls and travelling waves||
|| Spectral stability analysis | (ii) Identify secondary flows that bifurcate from the neutral curves determined in i) | 0.01-7 | 0°-90° focusing on 89° |Identify individual 2D and 3D secondary states at specific values of the control parameters to find bicritical transitions between states.|
|| Spectral stability analysis | (iii) Linear stability analysis of the secondary flow | 0.01-7 | 0°-90° focusing on 89° | Test the stability of the states found by applying disturbances to the states found in ii. The growth rate of the leading eigenvalues will be used indicate the bifurcation points and stability boundaries via interpolation.||
|| Spectral stability analysis | (iv) Identify tertiary flows that evolve from the bifurcation points | 0.01-7 | 0°-90° | Identify individual 3D states that form at or just beyond the bifurcation points and stability boundaries.||
|| Spectral stability analysis | (v) Linear stability analysis of the tertiary flow | 0.01-7 | 0°-90° | Apply linear stability analysis described in (iii) to the tertiary states that bifurcate from the secondary states.||
|| Computational fluid dynamics | (vi) Concurrent modelling of transition in shear flows with commercial solvers | a) Determine the conditions that produce directly comparable results (stable) from both finite volume and spectral codes.||
|| Computational fluid dynamics | (vi) Concurrent modelling of transition in shear flows with commercial solvers | b) Study mixed convection for volumetrically heated flows, where the velocity scale is coupled to the temperature field. ||
|| Computational fluid dynamics | (vi) Concurrent modelling of transition in shear flows with commercial solvers | c) Establish criteria for the onset of mixed convection, recirculation and turbulence to provide a direct control criterion. ||
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• In using computational fluid dynamics, we divided the fluid layer into a large number of fluid elements in which we could calculate the local variation in the pressure, temperature and velocity as we applied a volumetric heat source. Care was taken to ensure that the number of elements used could capture the phenomena, which should correspond to the states observed in the spectral stability analysis.
Results
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Tables 2 and 3 give an overview of the work packages completed during the course of the fellowship. The complexity of the state space found using the spectral stability analysis for conditions of constant pressure gradient lead us to decide to concentrate on these conditions. It allowed us to gain as good as possible understanding of the state space at the transition from laminar conduction to laminar convection to unsteady coherent wavy flow as possible.
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Table 2: Overview of work completed – horizontal layer.
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||Numerical Method | Work Packages ||
|| - | For Conditions of Constant Pressure Gradient| Fluids (Pr) | Angle (χ) | Determine states and stable/unstable regions ||
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|| Spectral stability analysis | (i) Linear stability analysis of the basic flow | 0.025 0.25 0.705 and 7 | 0° | Determined the linear neutral curve (red curves in each plot in Fig. 3) at the transition from laminar state to hexagons and rolls. ||
|| Spectral stability analysis | (ii) Identify secondary flows that bifurcate from the neutral curves determined in i) | 0.025 0.25 0.705 and 7 | 0° | Identified individual 2D and 3D secondary states at specific values of the control parameters. ||
|| Spectral stability analysis | (iii) Linear stability analysis of the secondary flow | 0.025 0.25 0.705 and 7 | 0° | Tested the stability of the states found by applying disturbances to the roll states found in ii. Stability boundaries for various instabilities were found (regions above the linear neutral curves in Fig. 3). Stability of hexagonal states at around transition was also tested. ||
|| Spectral stability analysis | (iv) Identify tertiary flows that evolve from the bifurcation points | 0.025 0.25 0.705 and 7 | 0° | A number of tertiary states were found close to the stability boundaries in (iii). ||
|| Spectral stability analysis | (v) Linear stability analysis of the tertiary flow | 0.025 0.705 and 7 | 0° | Of the tertiary states found, none were stable and therefore not reportable. ||
|| Computational fluid dynamics | (vi) Concurrent modelling of transition in shear flows with commercial solvers | a) Conditions selected include the fluids described by Pr = (0.005 0.025 0.705 7). Here we modeled a horizontal fluid layer of infinite extent with an conducting boundary above and insulating boundary. The down-welling hexagons for Pr>0.5 and rolls below this value could be produced. An extended period of time was used to find a mesh that produced a hexagonal state that produced comparable wavenumbers.
Similar layers with closed boundary conditions were also modeled. ||
| (vi) Concurrent modelling of transition in shear flows with commercial solvers | b) & c) Mixed convection was not studied due to time constraints and delays in completing a) ||
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Table 3: Overview of work completed – inclined layer.
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||Numerical Method | Work Packages ||
|| - | For Conditions of Constant Pressure Gradient| Fluids (Pr) | Angle (χ) | Determine states and stable/unstable regions ||
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|| Spectral stability analysis | (i) Linear stability analysis of the basic flow | 0.025 0.705 and 7 | TR:
0°-8°; LR: 0°-90° ; HX: 0°-15° ; SQ: 0°-11° | Determined the bifurcation of the transition from laminar state to hexagons (HX), longitudinal rolls (LR) squares (SQ) and transverse rolls (TR) at the critical wavenumbers for the horizontal case for several different pressure gradients (Reynolds numbers of 0, 5, 10, 20 and 50) ||
|| Spectral stability analysis | (ii) Identify secondary flows that bifurcate from the neutral curves determined in i) | 0.025 0.705 and 7 | [2]: 0°-1.6° at 1.01Ra_c; Fellow: HXs 0°-16° and LRs 0°-40° for Ra_c-1.3Ra_c | Identified individual 2D and 3D secondary states at specific values of the control parameters including a zero pressure gradient over the range of volumetric Ra_c-1.3Ra_c. |
|| Spectral stability analysis | (iii) Linear stability analysis of the secondary flow | 0.025 0.705 and 7 | As (ii) | Tested the stability of the states found by applying disturbances to the roll states found in ii over the range of volumetric Ra_c -1.3Ra_c. ||
|| Spectral stability analysis | (iv) Identify tertiary flows that evolve from the bifurcation points | - | - | Tertiary states were time-dependent in nature, which required a DNS form of the spectral code to model them. ||
|| Spectral stability analysis | (v) Linear stability analysis of the tertiary flow | - | - | Tertiary states were time-dependent in nature, which required a DNS form of the spectral code to model them. ||
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• Spectral stability analysis (Work package i-v)
• We examined the state space stability for fluids with Prandtl numbers over the range 0.025-7. Over this range of Prandtl number there is a change in the structures formed as there is an exchange in the stability of the up and down hexagons (Fig. 2). This occurs at a Prandtl number of ~0.25 [13], [17]-[18]. At around this point rolls become the most stable planform. Only weakly non-linear and direct numerical simulations have been previously used in similar studies [13], [17]-[18]. These studies confirmed their findings with slight discrepancies in the regions of the stability of rolls compared to either hexagon type [1].
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• The stability of the rolls was examined in more detail at specific Prandtl numbers corresponding to liquid mercury, air and water (Pr = 0.025 0.705 and 7 respectively) as well as Pr=0.25 [1]. A number of steps were used to find the state space regions of the stability of different flow states. The transition between laminar flow and convection (thick red lines in Fig. 3) using linear analysis was first determined. Then rolls and at many different levels of heating (defined in the Fig. 3 in terms of the ratio of the Rayleigh number to its value at the critical transition between laminar conduction and buoyant convection, Ra_c) and the wavenumber were found within this linear state space region. Numerous disturbances were applied to these states to find the boundaries of the regions where the rolls are stable (the shaded regions in Fig. 3). The disturbances included hexagons, as they are known to be stable states in the same regions, while Eckhaus, cross-roll, knot, skewed varicose and oscillatory “roll” states were observed. The size of the regions of the stable rolls increases with the Prandtl number, where mercury quickly transitions to unsteady convection with oscillatory rolls. Oscillatory states occur with Pr = 0.025 0.25 and = 0.705 while they were not observed for water. The findings reported in [1] are consistent with established literature in the form of Rayleigh-Bénard convection (heating of fluid layers from below) [6], [25]-[30].
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• Further studies performed include the analysing the effect that the angle of the layer or the application of the pressure gradient were explored. For the inclination of the fluid layer it was found that the hexagons only occur for a very shallow inclination (~0.1°) before rolls aligned perpendicularly to the direction the layer was inclined became the dominant planform. This resulted in oscillating and travelling waves, which would require direct numerical simulation to fully resolved [2]. These studies were only performed at a single heating value. The Fellow extended the state space regions resolved by considered more internal heating values and larger angles (see Table 4), which requires further analysis. Note that in these studies only rolls (RL) and hexagons (HX) were examined. Two types of hexagon were considered and these were down-welling hexagons (HXD) where the fluid moves down at the cell centre and up-welling hexagons (HXU) where the fluid moves up at the cell centre. Two further distinctions were considered for each state modelled and that relates to their orientation with regards to the inclination of the layer. States indicated as type 1 are orientated in parallel to the inclination of the layer and those as type 2 are perpendicular to the inclination.
• For the effect of Reynolds number on structures formed, only the region of state space that defined the transition between laminar conductive flow and convective was explored as this also required fully DNS calculations to correctly resolve each data point. These studies would require much more time that that available to complete a comprehensive analysis [2].
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Table 4: Angles observed for each fluid, Pr=(0.025,0.705,7) at different levels of heating (Ra).
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|| Pr = 0.025 | Angle (°) ||
|| Ra/Ra_c | HXD type 1 | HXD type 2 | HXU type 1 | HXU type 1 | RL type 1 | RL type 2 ||
|| 1.004 | 0.190 | 0.220 | 0.395 | 0.240 | 0.350 | 5.25 ||
|| 1.010 | 0.290 | 0.340 | 0.585 | 0.145 | 0.535 | 8.05 ||
|| 1.016 | 0.360 | 0.425 | 0.685 | 0.095 | 0.670 | 10.10 ||
|| 1.022 | 0.420 | 0.490 | 0.855 | 0.070 | 0.780 | 11.50 ||
|| 1.027 | 0.470 | 0.550 | 0.955 | 0.055 | 0.875 | 13.20 ||
|| 1.033 | 0.520 | 0.605 | 1.050 | 0.055 | 0.960 | 14.50 ||
|| 1.039 | 0.565 | 0.665 | 1.130 | 0.065 | 1.040 | 15.70 ||
|| 1.053 | 0.655 | 0.555 | 1.310 | 0.065 | 1.215 | 18.30 ||
|| 1.068 | 0.735 | 0.560 | 0.410 | 0.560 | 1.385 | 20.50 ||
|| 1.082 | 0.805 | 0.570 | 0.370 | 0.570 | 1.550 | 22.40 ||
|| 1.097 | 0.870 | 0.585 | 0.370 | 0.585 | 1.700 | 24.20 ||
|| 1.111 | 0.930 | 0.600 | 0.185 | 0.145 | 1.850 | 25.80 ||
|| Pr = 0.705 | Angle (°) ||
|| Ra/Ra_c | HXD type 1 | HXD type 2 | HXU type 1 | HXU type 1 | RL type 1 | RL type 2 ||
|| 1.001 | 0.74 | 0.79 | 0.45 | 0.50 | 0.70 | 2.40 ||
|| 1.009 | 2.00 | 1.06 | 0.92 | 1.65 | 2.20 | 7.60 ||
|| 1.017 | 3.40 | 1.03 | 1.65 | 2.25 | 3.05 | 10.50 ||
|| 1.025 | 4.40 | 0.91 | 2.00 | 2.75 | 3.65 | 12.70 ||
|| 1.033 | 5.10 | 0.82 | 5.10 | 3.10 | 4.20 | 14.60 ||
|| 1.042 | 5.70 | 0.75 | 2.15 | 3.45 | 4.65 | 16.20 ||
|| 1.050 | 6.20 | 0.68 | 2.70 | 3.75 | 5.05 | 17.70 ||
|| 1.058 | 6.65 | 0.62 | 2.90 | 4.05 | 5.40 | 19.00 ||
||Pr = 7.000 | Angle (°) ||
|| Ra/Ra_c | HXD type 1 | HXD type 2 | HXU type 1 | HXU type 1 | RL type 1 | RL type 2 ||
|| 1.010 | 2.74 | 1.70 | 1.50 | 1.75 | 1.50 | 8.00 ||
|| 1.091 | 8.90 | 2.26 | 3.85 | 5.65 | 4.90 | 23.50 ||
|| 1.172 | 12.20 | 2.18 | 5.15 | 7.60 | 6.60 | 31.40 ||
|| 1.212 | 13.50 | 2.16 | 5.60 | 8.40 | 7.30 | 34.40 ||
|| 1.293 | 15.70 | 2.14 | 6.50 | 3.00 | 8.50 | 39.30 ||
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• Computational Fluid Dynamics (Work package vi)
• We describe the simulations performed on horizontal fluid layers which are open in the horizontal direction [3]-[5]. Closed systems were also considered where the vertical surfaces in the geometries considered consisted of adiabatic or insulating conditions [3]. They took the form of a horizontal layer and a vertical two dimensional square geometry [3].
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• The accuracy of the model setup was tested for the horizontal case with water as the convecting fluid against the critical wavenumbers [3]. The states predicted were accurate to 1 and 15%. These states observed took the form of hexagonal circulation cells as indicated in Fig. 5a-b, which correspond to states observed in [14]. Considering the wavenumber vector components the accuracy was between 3 and 16% for one component and 11 and 57% for the other at the transition to turbulence.
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• Simulations within the closed domain for the horizontal layer also show the formation of polygonal structures at the transition to convection (Fig. 6) based on the experimental studies [21]. Hexagons were observed in the centre of the larger domain (1:20:20), while only pentagons could be formed in the smaller domain (1:10:10). The larger of the two domains could be considered equivalent to the constant flux condition. The states observed in the larger domain are similar to those in the domains of infinite extent with the exception of the regions near to the vertical boundaries. Therefore, it was considered that repetition with the constant flux or closed boundary conditions would not result in significant differences for the states observed in horizontal layers using either spectral stability analysis or computational fluid dynamics.
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• Increases in the heating applied to the horizontal fluid layers with open boundaries on the vertical surfaces were simulated [4]-[5]. The structures presented elsewhere were obtained at Ra_c, 2 Ra_c, 3Ra_c, 6Ra_c and 12Ra_c. The structures observed at the higher heating levels were consistent with experimental results reported in [24].
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• Further simulations were performed for air, molten salts and liquid sodium [4]-[5]. From these simulations we found that the rolls were dominant at lower Prandtl numbers, as the region where down welling hexagons are stable becomes smaller as indicated by the change in stability of the down hexagons reported in [13], [17]-[18] and observed in the case of air (Fig. 8). Only rolls were obtained below Pr=0.50 and no up-welling hexagons were obtained below the exchange in stability indicated at Pr=0.25 by [13], [17]-[18].
• Vertical two-dimensional square (Fig. 9), which showed the potential effect of different Prandtl numbers had on the states produced [3]. The geometry considered corresponds to the experimental investigations reported in [10]. The structures were different from a single circulation cell for air, while two circulation cells were observed for water with the heat upwelling in the centre of the square.
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Industry Contact
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a) Dr Generalis with Dr Philip Davies (Aston University) and Charlie Patton of Seawater Greenhouses with the Fellow and Dr Takeshi Akinaga applied to the United Kingdom Department for International Development to model thermal convection and humidity in novel greenhouses. Dr Generalis with Dr Philip Davies also applied to the European Science Foundation for funding to host a workshop on thermal convection and humidity in greenhouses as a result of these contacts.
b) Dr Generalis with Dr Philip Davies (Aston University) and Dr Esam Elsarrag (Gulf Organisation for Research and Development, GORD) submitted a grant application to the Qatari National Research Foundation to model thermal convection and the transport of humidity in large open air buildings. GORD is a government organisation in Qatar promoting and assessing environmentally responsible buildings and construction practices in the Gulf region, including sustainable cooling and dehumidification.
Training courses, workshops and conferences
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a) Soft skills training provided by the staff development unit at the host
i) webpage development
ii) introductory course to e-learning aids (Blackboard and Blackboard-Collaborate)
iii) research student supervision
iv) tutoring small groups
v) doing well in interviews
b) Free external courses
i) EU-PRACE courses hosted by Edinburgh University (http://www.prace-ri.eu/(si apre in una nuova finestra)) which were Message-Passing Programming with MPI, Cray XE6 Performance Workshop and PGAS Programming with UPC and Fortran Coarrays (April 2012, November, 2012 and January 2013)
ii) An Interactive Fortran 90 Programming Course of Liverpool University (http://www.liv.ac.uk/HPC/HTMLFrontPageF90.html(si apre in una nuova finestra))
iv) Intel course on code optimisation and vectorisation hosted by Polyhedron Ltd and Manchester University (http://www.polyhedron.com/intel-seminar-manchester2012(si apre in una nuova finestra) November 2012)
v) Introduction to High Performance Applications Development (Intel R Xeon Processors & Intel R Xeon Phi TM Coprocessors, Webinars - February/March 2013)
vi) London Mathematical Society - Engineering and Physical Sciences Research Council Short Course (SC63) - Modern nonlinear PDEs in fluid dynamics (July 2013)
c) Conferences and workshops the Fellow attended
i) Newcomen conference on The Nuclear Industry, Museum of Science and Industry Manchester, June 2012
ii) Marie Curie Actions Conference, Dublin, July 2012
iii) Site Visit at the Culham Centre for Fusion Research, November 2012
iv) Streaks in Shear Flows Workshop, Imperial College, London, December 2012
v) Nuclear Research Seminar, Magdalene College, Cambridge, September 2013
d) Conferences and Workshops where the Fellow presented papers or gave talks
i) Presented a paper at the 5th International Conference on Chaotic Modeling, Simulation and Applications (CHAOS2012), Athens, June 2012 (http://www.cmsim.org/images/1_CHAOS2012_Proceedings_Papers_G-L.pdf(si apre in una nuova finestra))
ii) Gave a talk at Mathematical models in science and engineering , Aston University, March 2013.
iii) Presented a paper at the 15th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-15) May 2013
References
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[1] Cartland-Glover, G., Fujimura, K. & Generalis, S. (2014). In preparation.
[2] Seivwright, C. (2014) MSc. Thesis, Aston University, Birmingham.
[3] Cartland-Glover, G., Fujimura, K. & Generalis, S., (2013). 15th International Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-15), Pisa, Italy, 12-17th May 2013.
[4] Cartland-Glover, G., Fujimura, K. & Generalis, S. (2013). CMSIM J 2013(1), 19-30.
[5] Cartland-Glover, G., Fujimura, K. & Generalis, S. (2012). Proceedings, 5th Chaotic Modeling and Simulation International Conference, 12 – 15 June 2012, Athens Greece.
[6] D. L. Turcotte, J. Schubert and G. Schubert. “Geodynamics” second edition, pp. 244, 249, 266, & 285, Cambridge University Press, Cambridge, 2001.
[7] Busse, F. H. (2013). Leverhulme Trust Lecture: Turbulence. URL: http://www.youtube.com/watch?v=KlIlT6LnfAI(si apre in una nuova finestra) available at www.aston.ac.uk/eas/staff/a-z/dr-sotos-generalis/.
[8] Bartzanas, T., Fidaros, D., Baxevanou, C., & Kittas, C. (2012). 1st International Symposium on CFD Applications in Agriculture - ISHS Acta Horticulturae 1008, Valencia, Spain. http://www.actahort.org/books/1008/1008_24.htm(si apre in una nuova finestra).
[9] Asfia, F. J. & Dhir, V. K. (1996). Nucl. Eng. Des. 163, p333.
[10] Seung D. Lee, Jong K. Lee & Kune Y. Suh (2007). Nucl. Eng. Des. 237, p473.
[11] Briant, R. C. & Weinberg, A. M. (1957). Nucl. Sci. Eng. 2, p797.
[12] Roberts, P. H. (1967). J. Fluid Mech. 30, p33.
[13] Tveitereid, M. & Palm, E. (1976). J. Fluid Mech. 76, p481.
[14] Thirlby, R. (1970). J. Fluid Mech. 44, p673.
[15] Ichikawa, H., Kurita, K., Yamagishi Y. & Yanagisawa T. (2006). Phys. Fluids 18, 038101.
[16] Cartland-Glover, G. & Generalis, S. (2009). Eng. Appl. Computat. Fluid Mech. 3, p164.
[17] Kolmychkov, V. V., Mazhorova, O. S. & Shcheritsa, O. V. (2013). Phys. Lett. A 377(34-36), p2111.
[18] Busse, F. H. (2014) Eur. J. Mech. B/Fluids 47, 32-34.
[19] Tritton, D. J. & Zarraga, M. N. (1967). J. Fluid Mech. 30, p21.
[20] Schwiderski, E. W. & Schwab, H. J. A. (1971). J. Fluid Mech. 48(4), p707.
[21] Kulacki, F. A.& Goldstein, R. J. (1972). J. Fluid Mech. 55, 271.
[22] Carrigan, C. R. (1985). Geophys. Astrophys. Fluid Dynamics 32, p1.
[23] Tasaka, Y., Kudoh, Y., Takeda, Y. & Yanagisawa, T. (2005). J. Phys. Conf. Ser. 14, p168.
[24] Tasaka, Y. & Takeda, Y. (2005). Int. J. Heat Mass Tran. 48, p1164.
[25] Takahashi, J., Tasaka, Y., Murai, Y., Takeda, Y. & Yanagisawa, T. (2010). Int. J. Heat Mass Tran. 53, p1483.
[26] Busse, F. H. (1967). J. Fluid Mech. 30(4), p625.
[27] Busse, F. H.& Whitehead. J. A. (1971). J. Fluid Mech. 76(2), p305.
[28] Busse, F. H. (1978). Rep. Prog. Phys. 41(12), p1929.
[29] Busse, F. H. & Clever, R. M. (1979). J. Fluid Mech. 91(2), p319.
[30] Busse, F. H. & Bolton, E. W. (1984). J. Fluid Mech. 146(1), p115.
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PUBLISABLE FINAL REPORT – Marie Sklowdoska Curie Intra European Fellowship
T2T-VHF
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This report collates the results obtained during the Marie-Curie Fellowship entitled Transition to Turbulence in Volumetrically Heated Flows (T2T-VHF - Project Reference 274367). The results are also described in more detail in references [1]-[5].
Aims
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The transition from the uniform steady laminar flow to the chaotic motion of turbulent flow is of great importance in fluid dynamics. The main motivation behind the proposed research project is to study the pre-chaotic bifurcation behaviour of strongly non-linear equilibrium solutions for incompressible volumetrically heated shear flows (VHSF) in a long channel. In particular, it is envisaged that the solutions obtained will be related to the advanced stages of transition identified by the studies involving DNS and experimental investigations in the turbulent regime.
Background
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A volumetrically heated fluid is a fluid that is heated from within. This can be heating caused by chemical or biochemical reactions or radioactive decay of elements in the earth [6] or nuclear reactors. The heat release or absorbed as a fluid component changes its physical state (i.e. from vapour to liquid or vice versa) can also be considered as volumetric heating [6]. Therefore, the phenomena we consider has a multitude of examples, which include convection in clouds and weather patterns, the transport of humidity through greenhouses (Fig. 1a, [8]) and the heating or cooling of chemical or biochemical reactors. Modelling the impact that convection has on the growth of clouds, weather patterns and severe weather, could help in their longer term prediction to enable better planning for societal responses to such weather (i.e. from disaster response and warnings to longer term flood mitigation). In the growth of clouds and greenhouses changes to the density of air occur as local changes in temperature and the water vapour content can lead to the formation of stable or unstable convection states, which can in turn affect weather formation and the growth of plants in greenhouses.
Several important industrial examples include decay heat removal in nuclear reactors, the effect that water vapour and coolant sprays have on fluid motion in the containment vessels of nuclear reactors as well as the convection of molten reactor cores that occurs in lower reactor vessel head or in core-catchers (Fig. 1b, [9]-[10]). The convection promotes mixing and it has an impact on how heat from the hot reactor or molten core is transported away from them. Understanding the impact that convection has on the turbulent flows observed during these events will help in developing mitigation strategies in order to better control undesirable situations. Molten salt nuclear reactors are also considered as volumetrically heated fluids as the fuel is dissolved in the molten salt and it can influence the turbulence observed [11].
The flow states that form at the transition between laminar convection and steady convection in volumetrically heated flows in the horizontal configuration are planforms consisting of either rolls or hexagons [1]-[5],[12]-[30]. The hexagons can either be down-welling (i.e. fluid moves down in the centre of the circulation) or up-welling (i.e. fluid moves up in the circulation centre). The states observed depend upon the nature of the fluid considered for example liquid mercury will result in the formation of up-hexagons or rolls while air and water give down-welling hexagons or rolls.
Objectives
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In the proposed study, we intend to model the transition to turbulence of VHSF. Two modelling approaches will be used to characterise natural and mixed convection. The first most accurate approach that is used by researchers in convective flows is the spectral method. The second approach utilises finite volumes to discretise the required transport equations. The key difference between the spectral code and the finite volume methods is that only the spectral code, based on the Galerkin collocation method, can resolve the eigenvalues of the problem at hand. The key advantages of the first approach are that can easily identify stable and unstable flow states and it can do so very quickly. However, the key disadvantage is that the equations used in the spectral method must be representative of the geometries considered (i.e. flat plat, rotating cylinders, spheres etc), while many engineering problems have complex geometries (i.e. chemical reactors, pipe bundles in heat exchanger, nuclear reactor cores).
Deliverables
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What we intended to do here is identify states and conditions that can be modelled accurately by both numerical methods for direct comparison. In doing this, we intend to create a toolbox that should allow us to apply appropriate amendments to the finite element code such that the relevant flow states identified by the spectral method can be also be modelled by the finite element method. More detail on the methods used and the results are reported in the corresponding subsections below. This would allow us to apply such techniques to model flows in more complex geometries that are not currently possible with spectral methods.
Methods
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Following the work plan set out in the proposal (Table 1), we used two approaches to model the convective flow states. The techniques used were spectral stability analysis and computational fluid dynamics.
• The spectral analysis involves solving the transport equations as a continuous harmonic expansion of modes (Chebyshev polynomials in the spanwise direction and Fourier modes in the longitudinal and transverse directions) that can describe the spatial dynamics of the state analysis. The states formed were controlled by the Rayleigh number, Ra, a measure of the buoyant convection in the system and the wavenumber, a non-dimensional characterisation of the size of the circulation cells which inversely proportional to the wavelength. The Prandtl number, Pr, is the dimensionless ratio of viscous to thermal diffusivity, which is 0.025 for liquid mercury, 0.705 for air and 7 for water. Therefore, Pr can control the behaviour of the fluid modelled.
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Table 1: Overview of Work Packages described on pages 6 and 7 of the proposal.
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|| Numerical Method | Work Packages ||
|| - | For Conditions of Constant Pressure Gradient and Constant Flux | Fluids (Pr) | Angle (χ) | Determine states and stable/unstable regions ||
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|| Spectral stability analysis | (i) Linear stability analysis of the basic flow | 0.01-7 | 0°-90° |Transition from laminar state to hexagons, rolls and travelling waves||
|| Spectral stability analysis | (ii) Identify secondary flows that bifurcate from the neutral curves determined in i) | 0.01-7 | 0°-90° focusing on 89° |Identify individual 2D and 3D secondary states at specific values of the control parameters to find bicritical transitions between states.|
|| Spectral stability analysis | (iii) Linear stability analysis of the secondary flow | 0.01-7 | 0°-90° focusing on 89° | Test the stability of the states found by applying disturbances to the states found in ii. The growth rate of the leading eigenvalues will be used indicate the bifurcation points and stability boundaries via interpolation.||
|| Spectral stability analysis | (iv) Identify tertiary flows that evolve from the bifurcation points | 0.01-7 | 0°-90° | Identify individual 3D states that form at or just beyond the bifurcation points and stability boundaries.||
|| Spectral stability analysis | (v) Linear stability analysis of the tertiary flow | 0.01-7 | 0°-90° | Apply linear stability analysis described in (iii) to the tertiary states that bifurcate from the secondary states.||
|| Computational fluid dynamics | (vi) Concurrent modelling of transition in shear flows with commercial solvers | a) Determine the conditions that produce directly comparable results (stable) from both finite volume and spectral codes.||
|| Computational fluid dynamics | (vi) Concurrent modelling of transition in shear flows with commercial solvers | b) Study mixed convection for volumetrically heated flows, where the velocity scale is coupled to the temperature field. ||
|| Computational fluid dynamics | (vi) Concurrent modelling of transition in shear flows with commercial solvers | c) Establish criteria for the onset of mixed convection, recirculation and turbulence to provide a direct control criterion. ||
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• In using computational fluid dynamics, we divided the fluid layer into a large number of fluid elements in which we could calculate the local variation in the pressure, temperature and velocity as we applied a volumetric heat source. Care was taken to ensure that the number of elements used could capture the phenomena, which should correspond to the states observed in the spectral stability analysis.
Results
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Tables 2 and 3 give an overview of the work packages completed during the course of the fellowship. The complexity of the state space found using the spectral stability analysis for conditions of constant pressure gradient lead us to decide to concentrate on these conditions. It allowed us to gain as good as possible understanding of the state space at the transition from laminar conduction to laminar convection to unsteady coherent wavy flow as possible.
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Table 2: Overview of work completed – horizontal layer.
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||Numerical Method | Work Packages ||
|| - | For Conditions of Constant Pressure Gradient| Fluids (Pr) | Angle (χ) | Determine states and stable/unstable regions ||
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|| Spectral stability analysis | (i) Linear stability analysis of the basic flow | 0.025 0.25 0.705 and 7 | 0° | Determined the linear neutral curve (red curves in each plot in Fig. 3) at the transition from laminar state to hexagons and rolls. ||
|| Spectral stability analysis | (ii) Identify secondary flows that bifurcate from the neutral curves determined in i) | 0.025 0.25 0.705 and 7 | 0° | Identified individual 2D and 3D secondary states at specific values of the control parameters. ||
|| Spectral stability analysis | (iii) Linear stability analysis of the secondary flow | 0.025 0.25 0.705 and 7 | 0° | Tested the stability of the states found by applying disturbances to the roll states found in ii. Stability boundaries for various instabilities were found (regions above the linear neutral curves in Fig. 3). Stability of hexagonal states at around transition was also tested. ||
|| Spectral stability analysis | (iv) Identify tertiary flows that evolve from the bifurcation points | 0.025 0.25 0.705 and 7 | 0° | A number of tertiary states were found close to the stability boundaries in (iii). ||
|| Spectral stability analysis | (v) Linear stability analysis of the tertiary flow | 0.025 0.705 and 7 | 0° | Of the tertiary states found, none were stable and therefore not reportable. ||
|| Computational fluid dynamics | (vi) Concurrent modelling of transition in shear flows with commercial solvers | a) Conditions selected include the fluids described by Pr = (0.005 0.025 0.705 7). Here we modeled a horizontal fluid layer of infinite extent with an conducting boundary above and insulating boundary. The down-welling hexagons for Pr>0.5 and rolls below this value could be produced. An extended period of time was used to find a mesh that produced a hexagonal state that produced comparable wavenumbers.
Similar layers with closed boundary conditions were also modeled. ||
| (vi) Concurrent modelling of transition in shear flows with commercial solvers | b) & c) Mixed convection was not studied due to time constraints and delays in completing a) ||
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Table 3: Overview of work completed – inclined layer.
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||Numerical Method | Work Packages ||
|| - | For Conditions of Constant Pressure Gradient| Fluids (Pr) | Angle (χ) | Determine states and stable/unstable regions ||
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|| Spectral stability analysis | (i) Linear stability analysis of the basic flow | 0.025 0.705 and 7 | TR:
0°-8°; LR: 0°-90° ; HX: 0°-15° ; SQ: 0°-11° | Determined the bifurcation of the transition from laminar state to hexagons (HX), longitudinal rolls (LR) squares (SQ) and transverse rolls (TR) at the critical wavenumbers for the horizontal case for several different pressure gradients (Reynolds numbers of 0, 5, 10, 20 and 50) ||
|| Spectral stability analysis | (ii) Identify secondary flows that bifurcate from the neutral curves determined in i) | 0.025 0.705 and 7 | [2]: 0°-1.6° at 1.01Ra_c; Fellow: HXs 0°-16° and LRs 0°-40° for Ra_c-1.3Ra_c | Identified individual 2D and 3D secondary states at specific values of the control parameters including a zero pressure gradient over the range of volumetric Ra_c-1.3Ra_c. |
|| Spectral stability analysis | (iii) Linear stability analysis of the secondary flow | 0.025 0.705 and 7 | As (ii) | Tested the stability of the states found by applying disturbances to the roll states found in ii over the range of volumetric Ra_c -1.3Ra_c. ||
|| Spectral stability analysis | (iv) Identify tertiary flows that evolve from the bifurcation points | - | - | Tertiary states were time-dependent in nature, which required a DNS form of the spectral code to model them. ||
|| Spectral stability analysis | (v) Linear stability analysis of the tertiary flow | - | - | Tertiary states were time-dependent in nature, which required a DNS form of the spectral code to model them. ||
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• Spectral stability analysis (Work package i-v)
• We examined the state space stability for fluids with Prandtl numbers over the range 0.025-7. Over this range of Prandtl number there is a change in the structures formed as there is an exchange in the stability of the up and down hexagons (Fig. 2). This occurs at a Prandtl number of ~0.25 [13], [17]-[18]. At around this point rolls become the most stable planform. Only weakly non-linear and direct numerical simulations have been previously used in similar studies [13], [17]-[18]. These studies confirmed their findings with slight discrepancies in the regions of the stability of rolls compared to either hexagon type [1].
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• The stability of the rolls was examined in more detail at specific Prandtl numbers corresponding to liquid mercury, air and water (Pr = 0.025 0.705 and 7 respectively) as well as Pr=0.25 [1]. A number of steps were used to find the state space regions of the stability of different flow states. The transition between laminar flow and convection (thick red lines in Fig. 3) using linear analysis was first determined. Then rolls and at many different levels of heating (defined in the Fig. 3 in terms of the ratio of the Rayleigh number to its value at the critical transition between laminar conduction and buoyant convection, Ra_c) and the wavenumber were found within this linear state space region. Numerous disturbances were applied to these states to find the boundaries of the regions where the rolls are stable (the shaded regions in Fig. 3). The disturbances included hexagons, as they are known to be stable states in the same regions, while Eckhaus, cross-roll, knot, skewed varicose and oscillatory “roll” states were observed. The size of the regions of the stable rolls increases with the Prandtl number, where mercury quickly transitions to unsteady convection with oscillatory rolls. Oscillatory states occur with Pr = 0.025 0.25 and = 0.705 while they were not observed for water. The findings reported in [1] are consistent with established literature in the form of Rayleigh-Bénard convection (heating of fluid layers from below) [6], [25]-[30].
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• Further studies performed include the analysing the effect that the angle of the layer or the application of the pressure gradient were explored. For the inclination of the fluid layer it was found that the hexagons only occur for a very shallow inclination (~0.1°) before rolls aligned perpendicularly to the direction the layer was inclined became the dominant planform. This resulted in oscillating and travelling waves, which would require direct numerical simulation to fully resolved [2]. These studies were only performed at a single heating value. The Fellow extended the state space regions resolved by considered more internal heating values and larger angles (see Table 4), which requires further analysis. Note that in these studies only rolls (RL) and hexagons (HX) were examined. Two types of hexagon were considered and these were down-welling hexagons (HXD) where the fluid moves down at the cell centre and up-welling hexagons (HXU) where the fluid moves up at the cell centre. Two further distinctions were considered for each state modelled and that relates to their orientation with regards to the inclination of the layer. States indicated as type 1 are orientated in parallel to the inclination of the layer and those as type 2 are perpendicular to the inclination.
• For the effect of Reynolds number on structures formed, only the region of state space that defined the transition between laminar conductive flow and convective was explored as this also required fully DNS calculations to correctly resolve each data point. These studies would require much more time that that available to complete a comprehensive analysis [2].
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Table 4: Angles observed for each fluid, Pr=(0.025,0.705,7) at different levels of heating (Ra).
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|| Pr = 0.025 | Angle (°) ||
|| Ra/Ra_c | HXD type 1 | HXD type 2 | HXU type 1 | HXU type 1 | RL type 1 | RL type 2 ||
|| 1.004 | 0.190 | 0.220 | 0.395 | 0.240 | 0.350 | 5.25 ||
|| 1.010 | 0.290 | 0.340 | 0.585 | 0.145 | 0.535 | 8.05 ||
|| 1.016 | 0.360 | 0.425 | 0.685 | 0.095 | 0.670 | 10.10 ||
|| 1.022 | 0.420 | 0.490 | 0.855 | 0.070 | 0.780 | 11.50 ||
|| 1.027 | 0.470 | 0.550 | 0.955 | 0.055 | 0.875 | 13.20 ||
|| 1.033 | 0.520 | 0.605 | 1.050 | 0.055 | 0.960 | 14.50 ||
|| 1.039 | 0.565 | 0.665 | 1.130 | 0.065 | 1.040 | 15.70 ||
|| 1.053 | 0.655 | 0.555 | 1.310 | 0.065 | 1.215 | 18.30 ||
|| 1.068 | 0.735 | 0.560 | 0.410 | 0.560 | 1.385 | 20.50 ||
|| 1.082 | 0.805 | 0.570 | 0.370 | 0.570 | 1.550 | 22.40 ||
|| 1.097 | 0.870 | 0.585 | 0.370 | 0.585 | 1.700 | 24.20 ||
|| 1.111 | 0.930 | 0.600 | 0.185 | 0.145 | 1.850 | 25.80 ||
|| Pr = 0.705 | Angle (°) ||
|| Ra/Ra_c | HXD type 1 | HXD type 2 | HXU type 1 | HXU type 1 | RL type 1 | RL type 2 ||
|| 1.001 | 0.74 | 0.79 | 0.45 | 0.50 | 0.70 | 2.40 ||
|| 1.009 | 2.00 | 1.06 | 0.92 | 1.65 | 2.20 | 7.60 ||
|| 1.017 | 3.40 | 1.03 | 1.65 | 2.25 | 3.05 | 10.50 ||
|| 1.025 | 4.40 | 0.91 | 2.00 | 2.75 | 3.65 | 12.70 ||
|| 1.033 | 5.10 | 0.82 | 5.10 | 3.10 | 4.20 | 14.60 ||
|| 1.042 | 5.70 | 0.75 | 2.15 | 3.45 | 4.65 | 16.20 ||
|| 1.050 | 6.20 | 0.68 | 2.70 | 3.75 | 5.05 | 17.70 ||
|| 1.058 | 6.65 | 0.62 | 2.90 | 4.05 | 5.40 | 19.00 ||
||Pr = 7.000 | Angle (°) ||
|| Ra/Ra_c | HXD type 1 | HXD type 2 | HXU type 1 | HXU type 1 | RL type 1 | RL type 2 ||
|| 1.010 | 2.74 | 1.70 | 1.50 | 1.75 | 1.50 | 8.00 ||
|| 1.091 | 8.90 | 2.26 | 3.85 | 5.65 | 4.90 | 23.50 ||
|| 1.172 | 12.20 | 2.18 | 5.15 | 7.60 | 6.60 | 31.40 ||
|| 1.212 | 13.50 | 2.16 | 5.60 | 8.40 | 7.30 | 34.40 ||
|| 1.293 | 15.70 | 2.14 | 6.50 | 3.00 | 8.50 | 39.30 ||
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• Computational Fluid Dynamics (Work package vi)
• We describe the simulations performed on horizontal fluid layers which are open in the horizontal direction [3]-[5]. Closed systems were also considered where the vertical surfaces in the geometries considered consisted of adiabatic or insulating conditions [3]. They took the form of a horizontal layer and a vertical two dimensional square geometry [3].
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• The accuracy of the model setup was tested for the horizontal case with water as the convecting fluid against the critical wavenumbers [3]. The states predicted were accurate to 1 and 15%. These states observed took the form of hexagonal circulation cells as indicated in Fig. 5a-b, which correspond to states observed in [14]. Considering the wavenumber vector components the accuracy was between 3 and 16% for one component and 11 and 57% for the other at the transition to turbulence.
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• Simulations within the closed domain for the horizontal layer also show the formation of polygonal structures at the transition to convection (Fig. 6) based on the experimental studies [21]. Hexagons were observed in the centre of the larger domain (1:20:20), while only pentagons could be formed in the smaller domain (1:10:10). The larger of the two domains could be considered equivalent to the constant flux condition. The states observed in the larger domain are similar to those in the domains of infinite extent with the exception of the regions near to the vertical boundaries. Therefore, it was considered that repetition with the constant flux or closed boundary conditions would not result in significant differences for the states observed in horizontal layers using either spectral stability analysis or computational fluid dynamics.
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• Increases in the heating applied to the horizontal fluid layers with open boundaries on the vertical surfaces were simulated [4]-[5]. The structures presented elsewhere were obtained at Ra_c, 2 Ra_c, 3Ra_c, 6Ra_c and 12Ra_c. The structures observed at the higher heating levels were consistent with experimental results reported in [24].
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• Further simulations were performed for air, molten salts and liquid sodium [4]-[5]. From these simulations we found that the rolls were dominant at lower Prandtl numbers, as the region where down welling hexagons are stable becomes smaller as indicated by the change in stability of the down hexagons reported in [13], [17]-[18] and observed in the case of air (Fig. 8). Only rolls were obtained below Pr=0.50 and no up-welling hexagons were obtained below the exchange in stability indicated at Pr=0.25 by [13], [17]-[18].
• Vertical two-dimensional square (Fig. 9), which showed the potential effect of different Prandtl numbers had on the states produced [3]. The geometry considered corresponds to the experimental investigations reported in [10]. The structures were different from a single circulation cell for air, while two circulation cells were observed for water with the heat upwelling in the centre of the square.
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Industry Contact
----------------
a) Dr Generalis with Dr Philip Davies (Aston University) and Charlie Patton of Seawater Greenhouses with the Fellow and Dr Takeshi Akinaga applied to the United Kingdom Department for International Development to model thermal convection and humidity in novel greenhouses. Dr Generalis with Dr Philip Davies also applied to the European Science Foundation for funding to host a workshop on thermal convection and humidity in greenhouses as a result of these contacts.
b) Dr Generalis with Dr Philip Davies (Aston University) and Dr Esam Elsarrag (Gulf Organisation for Research and Development, GORD) submitted a grant application to the Qatari National Research Foundation to model thermal convection and the transport of humidity in large open air buildings. GORD is a government organisation in Qatar promoting and assessing environmentally responsible buildings and construction practices in the Gulf region, including sustainable cooling and dehumidification.
Training courses, workshops and conferences
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a) Soft skills training provided by the staff development unit at the host
i) webpage development
ii) introductory course to e-learning aids (Blackboard and Blackboard-Collaborate)
iii) research student supervision
iv) tutoring small groups
v) doing well in interviews
b) Free external courses
i) EU-PRACE courses hosted by Edinburgh University (http://www.prace-ri.eu/(si apre in una nuova finestra)) which were Message-Passing Programming with MPI, Cray XE6 Performance Workshop and PGAS Programming with UPC and Fortran Coarrays (April 2012, November, 2012 and January 2013)
ii) An Interactive Fortran 90 Programming Course of Liverpool University (http://www.liv.ac.uk/HPC/HTMLFrontPageF90.html(si apre in una nuova finestra))
iv) Intel course on code optimisation and vectorisation hosted by Polyhedron Ltd and Manchester University (http://www.polyhedron.com/intel-seminar-manchester2012(si apre in una nuova finestra) November 2012)
v) Introduction to High Performance Applications Development (Intel R Xeon Processors & Intel R Xeon Phi TM Coprocessors, Webinars - February/March 2013)
vi) London Mathematical Society - Engineering and Physical Sciences Research Council Short Course (SC63) - Modern nonlinear PDEs in fluid dynamics (July 2013)
c) Conferences and workshops the Fellow attended
i) Newcomen conference on The Nuclear Industry, Museum of Science and Industry Manchester, June 2012
ii) Marie Curie Actions Conference, Dublin, July 2012
iii) Site Visit at the Culham Centre for Fusion Research, November 2012
iv) Streaks in Shear Flows Workshop, Imperial College, London, December 2012
v) Nuclear Research Seminar, Magdalene College, Cambridge, September 2013
d) Conferences and Workshops where the Fellow presented papers or gave talks
i) Presented a paper at the 5th International Conference on Chaotic Modeling, Simulation and Applications (CHAOS2012), Athens, June 2012 (http://www.cmsim.org/images/1_CHAOS2012_Proceedings_Papers_G-L.pdf(si apre in una nuova finestra))
ii) Gave a talk at Mathematical models in science and engineering , Aston University, March 2013.
iii) Presented a paper at the 15th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-15) May 2013
References
----------
[1] Cartland-Glover, G., Fujimura, K. & Generalis, S. (2014). In preparation.
[2] Seivwright, C. (2014) MSc. Thesis, Aston University, Birmingham.
[3] Cartland-Glover, G., Fujimura, K. & Generalis, S., (2013). 15th International Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-15), Pisa, Italy, 12-17th May 2013.
[4] Cartland-Glover, G., Fujimura, K. & Generalis, S. (2013). CMSIM J 2013(1), 19-30.
[5] Cartland-Glover, G., Fujimura, K. & Generalis, S. (2012). Proceedings, 5th Chaotic Modeling and Simulation International Conference, 12 – 15 June 2012, Athens Greece.
[6] D. L. Turcotte, J. Schubert and G. Schubert. “Geodynamics” second edition, pp. 244, 249, 266, & 285, Cambridge University Press, Cambridge, 2001.
[7] Busse, F. H. (2013). Leverhulme Trust Lecture: Turbulence. URL: http://www.youtube.com/watch?v=KlIlT6LnfAI(si apre in una nuova finestra) available at www.aston.ac.uk/eas/staff/a-z/dr-sotos-generalis/.
[8] Bartzanas, T., Fidaros, D., Baxevanou, C., & Kittas, C. (2012). 1st International Symposium on CFD Applications in Agriculture - ISHS Acta Horticulturae 1008, Valencia, Spain. http://www.actahort.org/books/1008/1008_24.htm(si apre in una nuova finestra).
[9] Asfia, F. J. & Dhir, V. K. (1996). Nucl. Eng. Des. 163, p333.
[10] Seung D. Lee, Jong K. Lee & Kune Y. Suh (2007). Nucl. Eng. Des. 237, p473.
[11] Briant, R. C. & Weinberg, A. M. (1957). Nucl. Sci. Eng. 2, p797.
[12] Roberts, P. H. (1967). J. Fluid Mech. 30, p33.
[13] Tveitereid, M. & Palm, E. (1976). J. Fluid Mech. 76, p481.
[14] Thirlby, R. (1970). J. Fluid Mech. 44, p673.
[15] Ichikawa, H., Kurita, K., Yamagishi Y. & Yanagisawa T. (2006). Phys. Fluids 18, 038101.
[16] Cartland-Glover, G. & Generalis, S. (2009). Eng. Appl. Computat. Fluid Mech. 3, p164.
[17] Kolmychkov, V. V., Mazhorova, O. S. & Shcheritsa, O. V. (2013). Phys. Lett. A 377(34-36), p2111.
[18] Busse, F. H. (2014) Eur. J. Mech. B/Fluids 47, 32-34.
[19] Tritton, D. J. & Zarraga, M. N. (1967). J. Fluid Mech. 30, p21.
[20] Schwiderski, E. W. & Schwab, H. J. A. (1971). J. Fluid Mech. 48(4), p707.
[21] Kulacki, F. A.& Goldstein, R. J. (1972). J. Fluid Mech. 55, 271.
[22] Carrigan, C. R. (1985). Geophys. Astrophys. Fluid Dynamics 32, p1.
[23] Tasaka, Y., Kudoh, Y., Takeda, Y. & Yanagisawa, T. (2005). J. Phys. Conf. Ser. 14, p168.
[24] Tasaka, Y. & Takeda, Y. (2005). Int. J. Heat Mass Tran. 48, p1164.
[25] Takahashi, J., Tasaka, Y., Murai, Y., Takeda, Y. & Yanagisawa, T. (2010). Int. J. Heat Mass Tran. 53, p1483.
[26] Busse, F. H. (1967). J. Fluid Mech. 30(4), p625.
[27] Busse, F. H.& Whitehead. J. A. (1971). J. Fluid Mech. 76(2), p305.
[28] Busse, F. H. (1978). Rep. Prog. Phys. 41(12), p1929.
[29] Busse, F. H. & Clever, R. M. (1979). J. Fluid Mech. 91(2), p319.
[30] Busse, F. H. & Bolton, E. W. (1984). J. Fluid Mech. 146(1), p115.