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Title:Modeliranje sevalnega toplotnega toka v sistemu Navier-Stokesovih enačb z metodo robnih elementov
Authors:Crnjac, Peter (Author)
Škerget, Leopold (Mentor) More about this mentor... New window
Files:.pdf DOK_Crnjac_Peter_2016.pdf (10,61 MB)
 
Language:Slovenian
Work type:Dissertation (m)
Typology:2.08 - Doctoral Dissertation
Organization:FGPA - Faculty of Civil Engineering, Transportation Engineering and Architecture
Abstract:V dinamiki tekočin se srečujemo z vsemi tremi fizikalno različnimi mehanizmi prenosa toplote: difuzijo, konvekcijo in s pojavom sevanja, ki je zapleten nelinearni proces prenosa toplote in dobi pomembno vlogo pri dovolj visokih temperaturah. V delu analiziramo vpliv toplotnega sevanja na skupni prenos toplote v newtonski tekočini, ki absorbira, emitira in izotropno sipa. V sevalno semitransparentnih snoveh, npr. v neizotermnih sistemih z visokimi temperaturami, je potrebno obravnavati tekočino kot stisljivo, kjer razlike v gostoti pomembno vplivajo na oblikovanje hitrostnega in temperaturnega polja. Vpliv sevanja na skupni prenos toplote izrazimo v energijski enačbi, kjer v gostoti nekonvektivnega toplotnega toka upoštevamo poleg difuzijskega še sevalni tok. To storimo tako, da izraz za divergenco vektorja gostote sevalnega toplotnega toka vključimo v energijsko enačbo kot izvorni sevalni člen. Enačba, s katero rešujemo sevalno temperaturno polje, je integrodiferencialna in predstavlja še vedno zelo resen problem v računalniški dinamiki tekočin. Delež toplotnega sevanja računamo z dvema sevalnima modeloma, ki ju stroka priporoča za optično goste snovi: z difuzijsko in sferično harmonično aproksimacijo. V difuzijski aproksimaciji analiziramo vpliv toplotnega sevanja z Rosselandovim modelom, kjer podamo gostoto toplotnega toka v gradientni obliki s temperaturno odvisnim koeficientom sevalne prevodnosti. Koeficient sevalne toplotne prevodnosti upoštevamo v energijski enačbi kot nelinearni izvorni člen. Ker v bližini stene sipanje ni izotropno, analiziramo interakcijo sevanja z difuzno steno. Natančnost difuzijske aproksimacije izboljšamo tako, da robni pogoj na steni modificiramo z vpeljavo Robinovega sevalnega robnega pogoja. V sferični harmonični aproksimaciji analiziramo vpliv toplotnega sevanja na konvektivni prenos toplote z zahtevnejšim P_1 sevalnim modelom. V modelu izrazimo gostoto vpadnega energijskega toka na danem mestu v sevalnem polju z nelinearno nehomogeno eliptično Helmholtzovo enačbo, ki jo kot vodilno enačbo P_1 aproksimacije dodamo ohranitvenim enačbam za maso, gibalno količino in energijo. Z Marshakovim robnim pogojem rešujemo enačbo iterativno kot vezani sistem z energijsko enačbo. Reševanje Navier-Stokesovih enačb povežemo z izbiro hitrostno-vrtinčne formulacije in z numerično ločitvijo kinematike in kinetike toka od tlačnega polja. Z robno-območno integralsko metodo transformiramo parcialne diferencialne enačbe v integralske. Za utežno funkcijo izberemo Greenovo osnovno rešitev; tako z robno diskretizacijo v celoti zajamemo linearni del rešitve, medtem ko z delitvijo območja na celice upoštevamo nelinearne prispevke prenosnega pojava. Dobljeni sistem algebrajskih enačb zapišemo v matrični obliki z vektorjem vrednosti funkcije polja in normalnega odvoda na robu ter z vektorjem znanih vrednosti. Na koncu podamo rezultate testiranj pri modeliranju sevalnega toplotnega toka v sistemu Navier-Stokesovih enačb in učinkovitost robno-območne integralske metode ter pripadajoče numerične sheme. V vseh numeričnih primerih analiziramo problem skupnega prenosa toplote v zaprti kvadratni kotanji napolnjeni z zrakom, ki ga obravnavamo kot idealni plin. Hitrostno in temperaturno polje ter skupni prenos toplote izračunamo za Rayleighova števila v laminarnem režimu, rešitve pa primerjamo z objavljenimi standardnimi rezultati.
Keywords:računalniška dinamika tekočin, hitrostno-vrtinčna formulacija, naravna konvekcija, sevanje, prenosna sevalna enačba, robno-območna integralska metoda
Year of publishing:2016
Publisher:[P. Crnjac]
Source:Maribor
UDC:[536.22:532.5]:519.62/.64(043.3)
COBISS_ID:19591702 Link is opened in a new window
NUK URN:URN:SI:UM:DK:YFTE5PJX
Views:975
Downloads:145
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Secondary language

Language:English
Title:Radiative Heat Flux Modelling in the System of Navier-Stokeʼs Equations with The Boundary Element Method
Abstract:In fluid dynamics all three physically different mechanisms of heat transfer can occur: conduction, convection and the radiation phenomenon, which is a complex nonlinear mode of heat transfer gaining importance at sufficiently high temperatures. The objective of this work is to develop a numerical model to solve coupled problems involving conductive, convective and radiative heat transfer in an absorbing, emitting, and isotropic scattering medium. At temperatures which are high enough in radiative semitransparent fluids, it is necessary to treat them as compressible, where the difference in mass density has significant influence on forming the velocity and temperature field. The radiation impact on overall heat transfer is conveyed in the energy equation, where in the non-convective energy flux, besides the diffusion heat flux also the radiative flux needs to be taken into consideration. This is done in such a way that we include the term for the divergence of the radiative flux vector into the energy equation as the radiative energy source. The equation used to solve the temperature field is an integro-differential equation and still presents a serious issue in the computational fluid dynamics. In this study the contributions from radiant energy transfer are presented using two approaches for optical thick fluids: the Rosseland diffusion approximation with two different techniques for implementing boundary conditions and the P_1 approximation. In diffusion approximation we consider an absorbing and emitting medium with isotropic scattering. The impact of heat radiation is analysed by applying the Rosseland model, where the radiative flux is given in a gradient form with a temperature dependent coefficient of radiative conductivity. The coefficient of radiative conductivity is taken into account in the energy equation as a non-linear source term. However near a boundary the diffusion approximation may not be accurate as the radiation is not isotropic. To overcome this difficulty, the boundary condition at the edge of the medium is modified by applying the Robin radiative boundary condition. In the spherical harmonics approximation, the impact of radiation on the overall heat transfer is analysed by applying a more complicated P_1 radiative model. In the model, we express the incident radiation at a given position in the radiation field by the nonlinear nonhomogeneous modified Helmholtz equation. Under the Marshak boundary condition, we solve the equation iteratively as a coupled system with the energy equation. The governing equations are in the next step transformed with the use of the velocity-vorticity variables formulation into transport equations for kinematics and kinetics. By applying the boundary domain integral method, we transform partial differential equations into integral equations. Green's function is chosen for the weighting function; with the boundary discretization we obtain the linear part of the solution, whereas by partitioning the domain into internal cells, we take into account the nonlinear contributions of the transport phenomena. We note down the obtained system of algebraic equations in the matrix form with the column vector of unknown boundary field function and its normal derivative and with the vector of known values. To test the validity and accuracy of the proposed numerical scheme, we analyse the problem of the overall heat transfer in a closed square cavity filled with air, treated as the ideal gas. The velocity and temperature fields together with the total heat transfer are calculated for the Rayleigh numbers in the laminar regime, and the solutions compared to the published standard results.
Keywords:Computational Fluid Dynamics, Velocity-Vorticity Formulation, Natural Convection, Radiation, Radiative Transfer Equation, Boundary-Domain Integral Method


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