Algorithm Application Of Singular Value Decomposition-generalized Finite Difference For Thermal Fluidstructure Interaction

Thermal fluid-structure interaction is very common in our daily life,industrial application and natural environment.Re-meshing is required when the thermal fluid-structure interaction is solved by traditional ALE methods.The immersed boundary method is popular,but has low accuracy when dealing with adiabatic boundary.The generalized finite-difference（GFD）with singular value decomposition（SVD）approximation does not require re-meshing when dealing with moving bodies,and has at least second-order accuracy at the fluid-solid interface.Therefore,the development of the SVD-GFD method for the thermal fluid-structure interaction can broaden the existing numerical methods,provide more stable and efficient numerical tools to investigate some thermal fluid-structure interaction problems,such as the thermal-fluid interaction involving high Prandtl number or complex immersed objects.Simultaneously,the investigation of thermal fluid-structure interaction not only enhances our understanding of nature,but also contributes to the development of industrial technologies,such as sedimentation of river sand,fluidization of solid in chemical reactors and drug delivery in blood.There are some deficiencies for existing CFD methods to solve the thermal fluid-structure interaction with complex immersed objects or high Prandtl number.Hence,the SVD-GFD method is extended to deal with the thermal fluid-structure interaction problems.The Boussinesq approximation is introduced to handle the thermal buoyancy and to couple the temperature and flow fields.SVD-GFD is a hybrid Cartesian-meshless method,and the governing equations are solved by the conventional finite difference scheme on Cartesian nodes and the GFD approximation on meshless nodes.Simultaneously,SVD-GFD is also used to couple the two discrete systems of Cartesian nodes and meshless nodes.A projection method is applied to solve N-S equation,and the buoyancy term is considered as the source term.And the convective heat transfer equation is solved by the second-order semi-implicit Crank–Nicolson scheme.The motion of the object is solved by second-order Heun method.Therefore,the SVD-GFD method for the thermal fluid-structure interaction has second-order accuracy in time and at least accuracy second-order in space,and high geometric adaptability.To test the capacity of the present SVD-GFD method for the simulation of the thermal flows with complex immersed objects,a series of simulations are conducted,such as forced convection and mixed convection around a circular cylinder,thermal flows involving moving boundary,2D and 3D thermal fluid-structure interaction.The results shows that the present SVD-GFD method offers great flexibility to handle complex immersed objects in thermal flows,while computational stability and accuracy are guaranteed.These numerical examples demonstrate that the SVD–GFD method is successfully extended to simulate the thermal-fluid interaction involving high Prandtl number or complex immersed objects.Most of the research on particle sedimentation focuses on spherical,cylindrical or square particles with simple geometric configurations.Then,the sedimentation of a torus at low Reynolds number（Re _T）is numerically studied by the SVD-GFD method.The influences of initial inclination angle（θ₀）and Re _T on the flow field and the torus motion are analyzed.The terminal state of torus is determined by Re _T but independent ofθ₀.The torus exhibits a kind of a zigzag motion and rotation during sedimentation.And this zigzag motion is more obvious at larger Re _T andθ₀.The torus reaches a stable state with zero inclined angle.The vortex street is likely to occur at higher Re _T andθ₀.The recirculation zone attached to the inner wall is longer at a higher Re _T,and its length initially approaches the maximum value and then decreases to the valley.The pressure plays the main role in the generation of lift-like force and drag,and is also the reason for the deceleration of the falling speed of torus.And this phenomenon becomes more obvious with the increases of Re _T andθ₀.After that,a numerical study of the sedimentation of two particles with different sizes is carried out.The influences of the Reynolds number（Re）and the particle size difference ratio（β）on the sedimentation of particles are analyzed.Depending on Re andβ,three different terminal states exist when the two-particle system settles in channel,namely the steady-state,the periodic oscillation and the period-doubling bifurcation.The steady-state includes steady-state I（SSI）which is not dominated by the inertial effect and steady-state II（SSII）which is dominated by the inertial effect.There are two types of periodic oscillation,which are identified as the periodic motion without wake（referred to as PMI）and the periodic motion with wake（referred to as PMII）.The limit cycles of PMI and PMII are counterclockwise and clockwise,respectively.The size of the limit cycle of PMI which occurs at 8≤Re≤9 and 12≤Re<70 decreases and increases with increasingβ,respectively.And the size of the limit cycle of PMII increases with increasingβ.The periodic-doubling bifurcation occurs at 14≤Re≤30,and its limit cycle has two branches.Research on the particle sedimentation at high Prandtl number（Pr）is relatively lacking.Hence,the sedimentation of a cold particle in non-isothermal fluid is numerically studied by the thermal SVD-GFD method.The influences of Grashof number（Gr）and Pr on the sedimentation of particle and heat transfer are analyzed.Three different sedimentation states exist when the cold particle reaches its terminal,including steady sedimentation along the centerline（regime A）,oscillation with small amplitude（regime B）and steady vertical sedimentation near the wall（regime C）.The heat transfer is dominated by Pr and enhances with increasing Pr.Gr has less influence on heat transfer,and this influence is amplified at high Pr.The final state of the cold particle likely falls in regime B when Pr increases.The change of particle motion with Pr at high Pr is different from that at low Pr.For example,the terminal Reynolds number and amplitude in regime B increase rapidly with increasing in low Pr but decrease slowly with increasing Pr in high Pr.Simultaneously,the results show that separation occurs far from the particle,and the separation point gradually gets closer to the particle with increasing Pr.