Mathematical modeling and two-dimensional analysis of thermo-fluid dynamics
around heated rotating cylinders with heat transfer by forced convection to low
Reynolds numbers
Modelagem Matemática e Análise Bidimensional da Termofluidodinâmica em
torno de Cilindros Rotativos Aquecidos com Tranferência de Calor por
Convecção Forçada para Baixos Números de Reynolds
DOI:10.34117/bjdv6n10-221
Recebimento dos originais: 08/09/2020 Aceitação para publicação: 09/10/2020
Rômulo Damasclin Chaves dos Santos
Doutor
Instituição: Universidade Estadual de Campinas (Postdoctoral Research)
Endereço: Cidade Universitária Zeferino Vaz - Barão Geraldo, Campinas - SP, 13083-970 E-mail: damasclin@gmail.com
ABSTRACT
In this present work is employed the Immersed Boundary Method coupled to Virtual Physical Model (IBM/VPM) for the two-dimensional analysis thermo-fluid dynamic and numerical simulations of incompressible flows. The heated stationary and rotating cylinders with forced convection use the Navier-Stokes equations, to know, mass conservation, momentum and energy to modeling the physical problem. The calculation of the forces exerted on the cylinder is realized using the VPM, which is based on conservation equations of linear momentum and energy. The numerical simulation of the fluid temperature at each instant of time, determined through auxiliary points distributed in Cartesian coordinates. The motivation is contributing with the computational implementation of the methodology mentioned using code developed in C++ for an experimental and numerical analysis, imposing different rates of rotation, comparing with an experimental results. Few are the studies about the thermal effects in combination with the aerodynamic coefficients and dimensionless numbers, for example, Strouhal, Nusselt, Péclet and Reynolds. The results obtained prove the efficiency of the method.
Keywords: Immersed Boundary Method, Virtual Physical Model, Stationary and Heated Rotating
Cylinder.
RESUMO
No presente trabalho é utilizado o Método de Fronteira Imersa acoplado ao Modelo Físico Virtual (MFI/MFV) para a análise bidimensional de termofluidos dinâmicos e simulações numéricas de escoamentos incompressíveis. Os cilindros aquecidos estacionários e rotativos com convecção forçada usam as equações de Navier-Stokes, para saber, conservação de massa, quantidade de movimento (momentum) e energia para modelar o problema físico. O cálculo das forças exercidas no cilindro é realizado por meio do MFV, que se baseia em questões de conservação do momento linear e da energia. A simulação numérica da temperatura do fluido na interface é igual à temperatura do sólido em cada instante de tempo, determinada através de pontos auxiliares distribuídos em coordenadas cartesianas. A motivação é contribuir com a implementação computacional da metodologia mencionada utilizando um código desenvolvido em C++ para análise experimental e numérica, impondo diferentes taxas de rotação, comparando com um resultado experimental.
Poucos são os estudos sobre os efeitos térmicos em combinação com os coeficientes aerodinâmicos e números adimensionais, por exemplo, Strouhal, Nusselt, Péclet e Reynolds. Os resultados obtidos comprovam a eficiência do método.
Palavras-chave: Método de Fronteira Imersa, Modelo Físico Virtual, Cilindro Rotativo
Estacionário e Aquecido.
1 INTRODUCTION
There have been great scientific, industrial and technological interest in studies of immersed body flows, which may be heated or not. The analysis of these flows and the existing phenomena, such as, vortex generation and vortex shedding, heat transfer and the forces exerted by the fluid have their importance applications in several areas, such as, mathematics, physics and engineering. Laboratory and numerical experiment ([2], [13]) have been done to obtain correlations with some dimensionless numbers, such as, Nusselt, Strouhal, Reynolds, velocity, pressure, vorticity and temperature fields, and still the relevant aerodynamic coefficients, to know, the lift, drag and pressure coefficients. Other works, such as, Chang and Finlayson [3] applied the finite-element method to study the heat transfer from a cylinder to a Newtonian fluid in laminar flow. Chuan-Chieh Liao and Chao-An Lin [4] checked the influence of the Prandtl number on the flux structure in different Rayleigh numbers in natural convection within domains with curved using the IBM.
In this work, the IBM is used to model the presence of immersed geometries in incompressible fluid around of the a heated stationary cylinder with and without rotation. Initially, the method was developed by Peskin [17] to study flow around heart valves. A term source of force
𝐹⃗ is introduced into the Navier-Stokes equations to model the solid-fluid interface. Analogously, in the same study, a heated body is modeled by term force of source 𝑄. Thus, the method is based on a mixed formulation with a mesh for the fluid (fixed Eulerian mesh) and another for fluid-solid interface (mobile/deformable – Lagrangian mesh). The employs two independent meshes, one for Eulerian variables in the fluid and other for the Lagrangian variables associated with the immersed boundary. The main difficulty in the spatial discretization consists in the construction of suitable approximation of the Dirac delta function which is used to take into account the interaction equations, see [18]. The problem is highly non-linear and presents several difficulties with the problem proposed using the Navier-Stokes equations.
In this work, the great interest is to verify the applicability and potential of Immersed Boundary Method (IBM) together with the Virtual Physical Model (VPM) for mathematical modeling and numerical simulation of flow over heated rotating cylinders. In other words, the interest is to analyze the heated transfer process in a heated cylinder, its effects during the rotational
movement, consider the drag, lift and dimensionless numbers, to know, Strouhal and Nusselt, stationary and rotating cylinders with forced convection for different values of the specific rotation (𝛼), comparing numerical results with experimental available in the literature. There are a variety of numerical studies that can be found in the literature involving thermo-fluid problems for a single cylinder. Some works, see [11, 12. 13, 14], are based on method IBM/VPM that has the objective to produce data and understand the dynamics of the flow. The key advantages the use of the IBM/VPM are low computational cost, easy mesh generation compared with other methods using unstructured meshes, ease of representing immersed bodies and calculate the forces acting on the surface the immersed body. Numerical simulations are performed in forced convection for different Reynolds numbers and different values of specific rotation (𝛼), as said before, with the aim of identify variation in aerodynamic coefficients, the Strouhal and Nusselt numbers.
Lima E Silva [11] developed a model called VPM, to calculate the interfacial force Lagrangian. The VPM is based on the Navier-Stokes equations, the calculated forces is included as a source term in the Navier-Stokes equations. The VPM has the ability to self-adjust to the flow once the force required for “bake” fluid particles near the interface is calculated automatically. The interfacial force is calculated in Lagrangian points and distributed to the Eulerian points neighbors, with the aid of a function type Gaussian. Lima E Silva [14] applied the IBM/VPM methodology flows with the presence of cylinders. Results are presented for both configurations where positioned in triangular shape and side-by-side. This method has shown good results in several simulations. The results confirmed the validity of the analysis method in the flow dynamics and their it influences the heat transfer process.
Other works, for example, Feng et al. [5] applied the immersed boundary method with a difference method to study the thermal convection in particulate flows. Lai and Peskin [9] developed a second-order scheme for the immersed boundary method for the simulation of flow with presence of stationary cylinders. The calculation of the force field is done explicitly using a function that depends on a constant rigid and displacement of interface points. Was used sufficiently of this constant to prevent the movement of the interface and thus represent an arbitrary solid body. Values were obtained to the aerodynamics coefficients, the Strouhal number and experimental data using the first and second-order scheme were obtained. It was concluded that using the second-order scheme numerical results were closer to the experimental results.
In the present work, a computational code, implemented, developer in language C++ is used to study various Reynolds values and the rotation of the cylinder with different rotation rates - (𝛼). The main factors, is the influence the dynamics of generation a shedding of vortices besides the
influence of heating, which are a relevant parameters. In we case, the rotation is determined only with the projection of tangential velocity obtained by the angular velocity imposed on 𝑥𝑘 and 𝑦𝑘
components of velocity in each Lagrangian point, as showing in Fig. 1.
Figure 1: Illustration of the projection of tangential velocity (a) and normal tangential vectors (b).
The remainder of this work is organized as follows: In the Section 2, presents the IBM methodology, aimed to study the flow around circular cylinder stationary with and without heating, presents the theorical foundation of the IBM methodology, formulation for the fluid and VPM, the velocity calculations process, pressure and temperature, definition of some dimensionless parameters for a cylinder immersed. In the Section3 are presents the numerical method used to solve the Navier-Stokes equations, continuity and energy. In the Section 4, are presents the results with the use of IBM/VPM in numerical simulations for the flow around a immersed interface. The numerical results are compared with the literature and in particular, with a experimental results obtained by Carvalho [2]. Finally, in Section 5, are presents the conclusions.
2 IMMERSED BOUDARY METHODOLOGY
Important physical phenomena in fluid mechanics can be describe by mathematical modeling, which consists of a set partial diffetential equations, frequently non-linear, know as conservation laws of fluid mechanics, they are: conservation momentum, conservation of mass and energy. These laws togheter model the effects of forces in fluid dynamics as the energy changes that occur in different regions. Thus, thanks to high-performace computers and numerical methods, the soluction of many problems in fluid mechanics have been posible.
In this work, the Immersed Boundary Method is used to model the presence of solid bodies immersed in a two-dimensional flow (2D) of an incompressible fluid. A force term (𝐹⃗ ) introduced into the Navier-Stokes equations is used to model the solid-fluid interface. Similarly, the heating of the immersed body is modeled by a force source term (𝑄). Thus, the methodology is based on Mixed
formulation with a mesh for fluid (fixed Eulerian mesh) and another mesh for fluid-solid interface (mobile/deformable – Lagrangian mesh), as shown in Fig. 2.
Figure 2: Representation of Eulerian and Lagrangian meshes to a body immersed with arbitrary
The flow simulations around of a stationary cylinder with heated and present different rotation rates of different Reynolds numbers is performed. The computer program called IBM/VPM is based on the explicit solution of the Navier-Stokes equations and Energy via Euler’s method. The spatial discretization is calculated by finite differences centered. For correction of pressure for solutions of linear systems is used the modified strongly implicit procedure proposed by [22]. The vorticity fields, temperature, pressure. Aerodynamics coefficients, Nusselt and Strouhal numbers, are obtained. The following is presented the formulation that describes the IBM/VPM used in this work.
2.1 MATHEMATICAL FORMULATION FOR THE FLUID
The domensionless governing equations are written with the following hypotheses to laminar flow and two-dimensional: an Newtonian fluid and incompressible with properties constant, the therm of buoyancy is based on the Boussinesq approximation not appear, bacause in the present paper only cases of forced convection are studies. The terms sources of the energy equation and quantity of momentum are based on IBM, the viscous dissipation and compression are negligible in the energy equation. The mass conservation equations, momentum and energy for incompressible flows of Newtonian fluids can be written in a dimensionless form
where 𝑉 is the dimensionless velocity of fluid, 𝑝 is the dimensionless pressure. 𝑅𝑒 is the Reynolds number defined as 𝑅𝑒 =𝜌0 𝑈 𝐿
. Here, 𝑈 is the characteristic velocity and 𝐿 is the
𝜇
characteristic length of flow field, 𝜇 is the viscosity of fluid, 𝜌 is the density of fluid. 𝑃𝑒 is the Péclet number defined as 𝑃𝑒 = 𝑅𝑒 × 𝑃𝑟; and 𝑃𝑟 is the Prandtl number which is defined as 𝑃𝑟 =𝑐𝑝 𝜇,
𝑘
where 𝑐𝑝 is the heat capacity and 𝑘 is the coefficient of thermal conductivity. The dimensionless
temperature is defined as 𝜃̅ = 𝜃̅−𝜃̅0 .
𝜃̅𝑏−𝜃̅0
The term 𝐹⃗ (𝑥 , 𝑡) is the Eulerian force field. This term source of force 𝐹⃗ (𝑥 , 𝑡) models the existence of the interface immersed in the flow. Just as the term source of force 𝐹⃗ (𝑥 , 𝑡) “sees” body in the flow is not null in Eulerian mesh near the Lagragian mesh. The term 𝑄(𝑥 , 𝑡) (Eulerian energy) is responsible for the flow “feel” the presence of the heated solid interface. Both the force term 𝐹⃗ (𝑥 , 𝑡) and 𝑄(𝑥 , 𝑡) are obtained with the aid of the Dirac’s delta function. The terms 𝐹⃗ (𝑥 , 𝑡) and 𝑄(𝑥 , 𝑡) of Eq. (2) and Eq. (3) model the interaction between the immersed boundary and the fluid by calculating the force field and energy in the region where the body is immersed. In the VPM developed by [11], there are no constants to be adjusted, as in the models proposed by [19] and [6], moreover, is not necessary to use highly sophisticated algorithms to interpolate variable between two meshes. This model can be representing the presence interface by equations of motion and energy at each point of the Lagragian mesh. The discretization of the Dirac’s delta function is not possible, needs to be replaced by a function type interpolation. This function has the objective of
exchange information between the two meshes (Eulerian and Lagrangian), pressure, velocity and energy.
The Eq. (4) show the discrete formulation of calculating the Eulerian force, with the use of this interpolation
where 𝑁 is the number of Lagrangian points, and ∆𝑠𝑘 is the discrete Lagrangian distance
between two points, shown inf Fig. 2. The term 𝐷𝑖,𝑗 is distribution function in this work, has
properties of a Gaussian function. The formulation used was proposed by [11]. The distribution function 𝐷𝑖,𝑗 represented by
where,
and still,
and 𝑟 represents 𝑥𝑘−𝑥𝑖
] or
𝑦𝑘−𝑦𝑖]. The term ℎ is the size of the Eulerian mesh, (𝑥 , 𝑦 ) the
[
ℎ [ ℎ 𝑖 𝑖
coordinates of Eulerian point 𝑥 of the domain. The forces 𝐹⃗ (𝑥 , 𝑡) and 𝑄(𝑥 , 𝑡) ha values null throughout of the domain, except for the neighborhood of the immersed interface where virtually
model the presence of heated immersed body, even the body has a complex geometry or being in motion.
3 THE VIRTUAL PHYSICAL MODEL
In the present work use an alternative model for calculating the density of the Eulerian force 𝐹⃗ (𝑥 , 𝑡) and the term source of the heat 𝑄(𝑥 , 𝑡). The model permits the calculation of 𝐹⃗ (𝑥 , 𝑡) base on fluid-solid interaction. The “virtual” term refers to the fact the no-slipping is modeled without direct imposition of the velocity on interface. The model dynamically evaluates the force that the fluid exerts on the solid surface immersed in the flow and heat exchange between them. The
Lagrangian force 𝑓 (𝑥 , 𝑡) and the heat source 𝑞(𝑥 , 𝑡) are evaluate separately, while for the thermal source, is used the dimensionless energy equation, showing the iteration between the particle-fluid and interface, as shown in the Fig. 3, considering all the terms of the Navier-Stokes equation.
Figure 3: Volume control located on a particle fluid Lagragian.
The principle of conservation of mass, of momentum and energy is applied to any fluid particle that makes up a flow. These conservation principles should also be applied to the fluid particles that are in contact with the fluid-solid interface. Thus, talking a particle illustrated by Fig. 3, where one interface is immersed passing results in the following formulation
𝑎
where, the terms of the right side of Eq. 9 are respectively called acceleration force (𝑓 ), inertial (𝑓 ), viscous force (𝑓 ) and pressure force (𝑓 ).
𝑖 𝑣 𝑝
The different parts of the Eq. (9) must be evaluate on the interface using the Eulerian velocity
𝑉 (𝑥 𝑘, 𝑡) and pressure 𝑝(𝑥 𝑘, 𝑡) field. The velocity and pressure derivatives, are computed using the
Eq. 1 and Eq. 2. For the heat source, the particle-fluid has the following Eq. 10
it is important to know that these terms are calculated interface points through interpolation of pressure fields, velocity and temperature, calculated in Eulerian mesh.
3.1 INTERPOLATION AND DERIVATIVES CALCULATION
The interpolation scheme for the horizontal velocity, over arbitrary point shown in Fig. 4. A square dotted is defined as being the region where the variables will be analyzed. The shaded are denotes the immersed body (cylinder). The horizontal velocity components, shown in Fig. 4, are interpolated over point 3, using the same model given by Eq. 6 and Eq. 7. Only components that are a shorter distance or equal to 2∆𝑥 from the point 3, is interpolated. This procedure is executed automatically by the distribution function model. Furthermore, the box, shown in Fig. 4 is used to reduce the time of computation process. Starting from each point (𝑥𝑘) interface, draw two parallel
lines to the system coordinate system axes, in the direction the exterior of exterior of interface. In each direction are marked two distant points ∆𝑥 and 2∆𝑥 interface.
Figure 4: Interpolation scheme for the horizontal (a) and vertical (b) at arbitrary point.
This distance is necessary to avoid that two auxiliary points are allocated inside one Eulerian volume. Meshes that are at a greater distance 2∆𝑥 the Lagrangian point, do not contribute to the interpolation. For each time step, express discretization is done by 𝑢𝑘−𝑢𝑓𝑘 and 𝑣𝑘−𝑣𝑓𝑘, where 𝑢 , 𝑣
∆𝑡 ∆𝑡 𝑘 𝑘
represent the interface velocities and 𝑢𝑓𝑘, 𝑣𝑓𝑘, the fluid velocities on the interface, considering the
interval velocity and the external interface on the Eulerian mesh.
Similarly, the fluid temperature at the interface is equal to the solid temperature in each instant of time. The general equation for the velocity at Lagrangian points is expressed by
Where 𝑉 (𝑥 𝑘) is the Lagrangian velocity calculated on auxiliary point (𝑥 𝑘) by interpolation
of Eulerian velocity 𝑉 (𝑥 𝑖). For the calculation of the derivative of pressure and temperature at each
point Lagrangian it is necessary to obtain the value of pressure and temperature on the surface at the point (𝑥 𝑘). To calculate the pressure and temperature was used an auxiliary point 𝑃, which is normal
in position in position at a distance ∆𝑥 of Lagrangian point. It is observed in Fig. 5, that the pressure and temperature in this auxiliary point 𝑃 belong to an Eulerian cell, both transported to the interface.
Figure 5: Illustration of the interpolation procedure for pressure (𝑝) (blue color) and temperature (𝜃̅) (red color).
The calculation process uses auxiliary points (1, 2, 3 and 4) for Fig. 4, is compute their derivatives in the 𝑥 and 𝑦 directions. The general equations for obtaining the pressure and temperature in points (1, 2, 3 and 4) are given by
where 𝑝(𝑥 𝑖) and 𝜃̅(𝑥 𝑖) are respectively the pressure and temperature in the Eulerian mesh to
be interpolated. The terms 𝑝(𝑥 𝑘) and 𝜃̅(𝑥 𝑘) are the pressure and temperature Lagrangian in Fig. 5.
The calculation is done using a second-order Lagrange polynomial approximation.
Generalizing the vertical and horizontal velocity components and the pressure by 𝜙, the first and second derivatives in the 𝑥 and 𝑦 directions are approximated by Eq. (14) and (15)
the derivatives in the 𝑦 direction is analogous, 𝜙𝑘 are obtained by interpolation describe in
Fig. 4. The same process is valid for energy equation.
4 NUMERICAL METHOD
The numerical method used in this paper is the fractional steps that unites the velocity and pressure. With the aim to solve the Navier-Stokes equations and the continuity equation, result new velocity and pressure fields. For the time discretization is used Euler’s method of the first-order. The Navier-Stokes equation were solved explicitly. The correction of pressure results in a linear system, solved by Modified Strongly Implicit Procedure, developed by [22]. The Eq. 2 of Navier- Stokes can be rewritten in the form indexical, in this case, the velocity for the current iteration is:
The approximation of the velocity components are made using the pressure fields, velocity and force, calculated in the previous iteration Eq. (16), is given by
where (𝑢 ) is the estimated velocity component, ∆𝑡 is the computational time step, 𝑛 is sub- step index.
Subtracting the Eq. (17) of the Eq. (18), result in
Organizing the terms, obtain
where 𝜕𝑝′𝑛+1 = 𝑝𝑛+1 − 𝑝𝑛 is the pressure correction. The calculation of the divergent is important because express the conservation of mass. For an incompressible fluid flow, the value of the divergence is used as a guarantee of the conservation all around the computational domain. With this assured condition, the second term on the left side of Eq. 21 is canceled. Thus, is rewrite as follows
Organizing the terms of Eq. (21), results in the Poisson equation for the correction of pressure, where the source term is the divergence of the approximate velocity
Therefore, of the Eq. (22), the velocity corrected is calculated for the current iteration, given by
5 RESULTS
Using the IBM/VPM, implemented in C++ code, is possible perform simulation of two- dimensional (2D) flows around a heated body immersed in the flow. The flow simulations around a stationary cylinder with rotation is performed to Reynolds number varying 47 ≤ 𝑅𝑒 ≤ 250. The cylinder is maintained at a constant temperature dimensionless equal to 1 (𝜃̅𝑐 = 1), while the fluid
has a initial temperature equal to 0 (𝜃̅𝑓 = 0). The value of the specific rotation (𝛼) is studied in the
Then, the results were compared to the literature, including the experiments conducted by [2]. The vorticity fields, temperature and pressure are shown, the lift and drag coefficients, the Strouhal and Nusselt numbers for different Reynolds numbers and specific rotations.
This work is restricted to simulate flow around a cylinder diameter 𝑑 and immersed in a heated incompressible fluid with constant properties. The simulations were performed with and without rotational movement of the cylinder, and it was possible to numerically validate the methodology and to analyze the influence of rotation on the thermal field. For all simulations, it is used a rectangular area of dimension (55𝑑 × 30𝑑) as shown in Fig. 6. These dimensions were determined numerically to minimize influences the flow field around the cylinder and at the same time, minimize the number unnecessary of nodes. The cylinder center coordinates are (16,5𝑑 × 15𝑑) in 𝑥 and 𝑦.
Figure 6: Schematic illustration of the dimensions of the calculation domain.
5.1 VISUALIZATION OF FLOW FIELD
The Fig. 7 show different time instants, the vorticity fields (providing a considerable sense of the movement of fluids), for Reynolds numbers equal to 100 and 200, respectively, around a stationary cylinder. The cylinder is maintained at a dimensionless constant temperature, i.e., (𝜃̅ = 1). The left column represents the different instants of time for the simulations with a specific rotation (𝛼 = 0), while the right column represents also the different time instants to flow around a rotating cylinder (𝛼 = −1.5) clockwise. The flow with stationary cylinder, the initial instants are marked by rise of recirculation bubble behind the cylinder. This region where the fluid is positioned constantly increases (𝑡 = 77𝑠) up to a maximum length. Then starts the rise shedding process of the first vortex. It is known that the process of generating and shedding vortices occurs due to the instabilities of the shear layers which around of geometry and the Reynolds number. It is observed
in Fig. 7 and Fig. 8 the cylinder stationary has an elongate wake, and the vortex shedding started after about a nondimensional time 𝑇 = 80.
Figure 7: Vorticity field for Re = 100 and Re = 200. The left column (a) and (b) show the stationary cylinder with 𝜶 = 0 and the rotating rate for 𝜶 = −1.5. In the right column (c) and (d) show the stationary cylinder for Reynolds Re = 200 and rotating rate 𝜶 = −1.5.
Figure 8: Qualitative comparison (a) and (b) between the experimental results by [2] left column. The right column and the numerical results of this present work to 𝑅𝑒 = 115 with different specific rotations.
The rotation anticipates the vortex shedding, destabilizing the flow, causing the wake of Von Kármán appear in the initial moments. For Reynolds value 100 and 200 with 𝛼 = −1.5, the vortex shedding is not inhibited and wake vortices is inclined downwards in the direction of rotation. The wake formed downstream has a similar inclination in both studies. The value of (𝛼) critical for 𝑅𝑒 = 115 is 𝛼 = 2.02, while the experimental was between 2.02 and 2.27.
5.2 RESULTS TO AERODYNAMICS COEFFICIENTS, STROUHAL AND NUSSELT NUMBERS
In the Tab. 1 are presents some results obtained in the present work for the average value (temporal means) of the drag coefficients, for Reynolds numbers equal to 80, 100 and 200 numerically simulated, compared with the numerical data of [20], [11] and [10], to case of (𝛼 = 0).
Table 1: Comparison of the average value of the drag coefficients for 80 ≤ 𝑅𝑒 ≤ 200.
Re Present Work Ding et al.
(2004) Lima E Silva (2002) Ye et al. (1999) Liu et al. (1998) 80 1.395 1.400 1.370 100 1.368 1.350 1.390 1.350 200 1.348 1.390 1.310
In Tab. 2 are the present other results obtained in the present work for the average Nusselt in comparison with some results obtained by [1], [15], [16] and [23].
Table 2: Comparison between results for 80 ≤ 𝑅𝑒 ≤ 200 and 𝑁 𝑢 , for 𝛼 = 0 .
Re Present Work Paramane (2009) Baranyi (2003) Mahir & Altac
(2008) Shrivastava et al. (2012) 80 4.6733 4.5000 100 5.1855 5.1320 5.1790 200 7.1625 7.4740 7.1600
It is observed a good approach of the values obtained, with differences of the order of 3%, which confirms the validity of the method for the case of the stationary cylinder.
6 CONCLUSIONS
The flow on circular cylinder with or without heating, with a rotary motion, have a potential application in several fields. The methodology used in this study is based on the application of conservation equation of momentum, mass and energy modeling the physics of flow. The motivation for the development of this work was to continue the work of [11], [12] and [13], seeking to contribute to the implementation of this methodology in flow problems on heating geometries subjected to forced convection. The computer code is used to simulate flow in the presence of stationary and rotating circular cylinder with heating. Results were obtained, such as vorticity field, temperature and pressure, and flow parameters characterizing the average values of the drag and lift coefficients, pressure, Strouhal and Nusselt numbers. These results were compared with experimental and numerical results, in order to validate the use of the methodology to the problem of moving boundary, validating the effect of the heat transfer by forced convection for Reynolds
numbers less than 250. With the main objective to better understand all the phenomena present in this flow, this work through a two-dimensional analysis of thermo-fluid dynamics, evaluate the thermal influence of the cylinder on the flow, the issuance of vortices and the dynamics of detachment of the Von Kármán wake, for Reynolds numbers of 47 ≤ 𝑅𝑒 ≤ 250, with rotating specific in the range of 0 < 𝛼 < 4.0.
It was found the influence of the rotating in reducing drag and increasing lift, and the distribution of the thermal field near the cylinder. Was obtained distribution of the pressure coefficient over the surface of the cylinder, the temporal evolution of some coefficients, the frequency of vortex shedding and the Nusselt number. It was observed an inclination with relation to increasing specific rotating. The velocity of the oscillations fluid dynamics of coefficients tends to null, i.e., the process of vortex shedding tends to decrease with increase specific rotating. Finally, it was found that the Strouhal numbers is little influenced for low values of the specific rotating, but is dependent on the Reynolds number. The quantitative results showed a good numerical convergence in relation to the results presented in the literature. The IBM/VPM methodology showed promising for all simulation.
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