Applications Numerical Simulation of Convection

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Numerical Heat Transfer, Part A: Applications

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Numerical Simulation of Convection-Diffusion Problems by the ControlVolume-Based Finite-Element Method

Odacir Almeida Nevesa; Estaner Claro Romãob; João Batista Campos Silvac; Luiz Felipe Mendes de Mourab a Federal Rural University of Semi Árido, DCEN, Mossoró, Brazil b State University of Campinas, FEM, DETF, Campinas, São Paulo, Brazil c São Paulo State University, DEM, Ilha Solteira, São Paulo, Brazil Online publication date: 04 June 2010

To cite this Article Neves, Odacir Almeida , Romão, Estaner Claro , Silva, João Batista Campos and de Moura, Luiz Felipe

Mendes(2010) 'Numerical Simulation of Convection-Diffusion Problems by the Control-Volume-Based Finite-Element Method', Numerical Heat Transfer, Part A: Applications, 57: 10, 730 — 748 To link to this Article: DOI: 10.1080/10407781003800672 URL: http://dx.doi.org/10.1080/10407781003800672

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Numerical Heat Transfer, Part A, 57: 730–748, 2010 Copyright # Taylor & Francis Group, LLC ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/10407781003800672

NUMERICAL SIMULATION OF CONVECTIONDIFFUSION PROBLEMS BY THE CONTROL-VOLUMEBASED FINITE-ELEMENT METHOD Odacir Almeida Neves1, Estaner Claro Romaõ2, Joaõ Batista Campos Silva3, and Luiz Felipe Mendes de Moura2 Federal Rural University of Semi A´rido, DCEN, Mossoro´, Brazil State University of Campinas, FEM, DETF, Campinas, Saõ Paulo, Brazil 3 Saõ Paulo State University, DEM, Ilha Solteira, Saõ Paulo, Brazil 1

Downloaded By: [UNICAMP] At: 19:11 30 June 2010

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This work presents a numerical study of the tri-dimensional convection-diffusion equation by the control-volume-based on finite-element method using quadratic hexahedral elements. Considering that the equation governing this problem in its main variable may represent several properties, including temperature, turbulent kinetic energy, viscous dissipation rate of the turbulent kinetic energy, specific dissipation rate of the turbulent kinetic energy, or even the concentration of a contaminant in a given medium, among others, the wide applicability of this problem is thus evidenced. Three cases of temperature distributions will be studied specifically in this work, in addition to one case of pollutant dispersion upon analysis of the concentration of a contaminant in a fixed flow point. Some comparisons will be carried out against works found in the open literature, while others will be done according to each phenomenon characteristics.

INTRODUCTION The area of computational simulation of fluid flows originated the branch of computational mechanics named computational fluid dynamics (CFD). Simulations using CFD have been increasingly used and accepted in projects so as to reduce the number of experiments required for the development of equipment and machines. In some cases, due to the great complexity, high risk or impossibility to carry out experiments with safety, it is the only tool available to predict the behavior of devices. A typical example of this kind of situation that poses risk to human health is the flow in the core of a nuclear reactor. Many projects have been optimized by using CFD. Reference [1] presents an excellent text on the utilization of CFD and its several aspects, such as modeling, construction of meshes, and solution of algebraic systems resulting from the application of computational techniques to partial differential equations that model flow and heat=mass transfer phenomena.

Received 6 November 2009; accepted 6 March 2010. Address correspondence to Luiz Felipe Mendes de Moura, Department of Thermal and Fluid Engineering, Universidade Estadual de Campinas, C. P. 6122, Campinas 13083-970, Brazil. E-mail: [email protected]

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SIMULATION OF CONVECTION-DIFFUSION PROBLEMS

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NOMENCLATURE A C C0 DC C Ce n Cab


J Ke Kab L Me Mab Na nj Pe Pr Prt Re RnU Sc Sct S/ SU S/e SUa t t ¼ t u0 =L T T0 DT u umax

area of the surface of a control volume dimensional concentration dimensional concentration of reference dimensional variation of concentration of reference dimensionless concentration convective matrix in the element coefficients of the matrix of convective terms Jacobian matrix diffusive matrix in the element coefficients of the matrix of diffusive terms length of reference mass matrix in the element coefficients of the mass matrix interpolation function component of the normal vector in direction of the xj axis Pe`clet number Prandtl number turbulent Prandtl number Reynolds number source term defined in Eq. (6b) Schmidt number turbulent Schmidt number source term in the transport equation of a scalar variable / dimensionless source term source term vector in the element coefficient of the vector of source terms dimensional time in seconds dimensionless time average or filtered temperature dimensional reference temperature dimensional variation of temperature of reference velocity in x-axis direction maximum velocity

ui Ui ¼ ui=u0 v V x xi Xi ¼ xi=L y w z a b h m mt q q / U C/ CU

dimensional velocity component in xi axis dimensionless velocity component in the dimensionless Xi axis velocity in y-axis direction volume of a control volume coordinate coordinates in the direction i, axis direction dimensionless coordinate in the direction i coordinate velocity in z-axis direction coordinate indication of a local node number or control volume indication of a local node number or interpolation function parameter indicating the time scheme discretization dynamic viscosity turbulent dynamic viscosity dimensional density dimensionless density dimensional scalar variable dimensionless scalar variable dimensional diffusion coefficient in the Eq. (1) dimensionless diffusion coefficient in the Eq. (3)

Superscripts n indication of time t level nþ1 indication of t þ Dt level indication of a dimensional variable Subscripts i a b 0

indication of direction of the axis or component of velocity indication of a local node number or control volume indication of local node number or interpolation function indication of a reference state

The computational fluid dynamics comprises the development and=or application of numerical methods to calculate the relevant magnitudes in flows at points of the physical domain, usually named nodal points or just nodes. The major methods used for the numerical simulations of fluid flows are the finite-difference method (FDM), finite-volume method (FVM), finite-element method (FEM), or their variants, such as control-volume-based on finite-difference method (CVFDM) and control-volume-based on finite-element method (CVFEM).


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O. A. NEVES ET AL.

All of these methods are actually derived from a single method, known as the weighted residual method, which is mathematically differentiated by the weighting function, or simply, the weight function applied in the residual vanishing. Some remarks on the major methods used for the calculation of fluid flows are provided below. The finite-difference method has been used to calculate fluid flows and heat transfer, which has been observed in the large number of works published. There are many computational codes based on this method. Several authors state that some kind of limitation to this method is in the discretization of domains with geometrical complexity, which is a problem that can be partially solved by using nonorthogonal meshes. A method presented in reference [2] and mentioned in the literature as the finite-volume method, also known by many authors as the control-volume-based finite-difference method, is the most used method for the numerical analysis of flows and heat transfer. The major characteristic of this method is the easy physical interpretation of the terms in the balance equations as terms of fluxes, sources, and forces. The finite-volume method with orthogonal and nonorthogonal meshes in generalized coordinates for dealing with irregular geometries has been implemented by several research groups and applied in the solution of flow and heat transfer equations [3]. The finite-element method, initially developed for the analysis of structures, has now been applied in the case of flows due to its great versatility in the discretization of geometrically complex domains. The method became widely accepted in the 1960s, when research was triggered in several locations worldwide. Since 1967, upon the insertion of the method, vast literature covering theory and application [4] is available. A number of basic references that address the application of the finiteelement method (FEM) in fluid flows include references [5–11]. The classical finite-element method is known as the Galerkin’s finite-element method (GFEM). Another variant of the finite-element method is known as leastsquares finite-element method (LSFEM). This work addresses the third variant of the finite-element method, known as the control-volume-based finite-element method (CVFEM) or also sub-domain finite-element method. This method was first presented by Baliga and Patankar [12, 13] and Baliga, Pham, and Patankar [14], by using triangular elements for domain discretization. Scheneider and Raw [15–17] presented this method for linear quadrilateral finite elements (4-node elements). Raw, Schneider, and Hassani [18] used a quadratic quadrilateral finite element (9-node element) for heat conducting transfers. An 8-node finite element (equivalent to the elimination of the central node in the previous element) known as serendipity element, has also been used in the Galerkin’s finite-element method to solve fluid flows. According to reference [9], CVFEM provides a combination of the FEM’s geometrical flexibility and the easy physical interpretation associated with the finite-volume method. The CVFEM formulation involves five basic steps [9]: 1) discretization of the element domain and further discretization in control volumes associated to the element nodes; 2) prescription of interpolation functions based on elements for dependent variables; 3) derivation of discretized equations by integration of the differential equations inside control volumes and substituting in the integrals the interpolation functions; 4) assembly of discretized equations element-by-element; and 5) prescription of a process or method to solve the resulting system of discretized equations.


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Campos–Silva [3] developed a computational code by using the sub-domain finite-element method, also known in the literature as the control-volume-based on finite-element method, for the numerical simulation of transient, viscous, and incompressible flows from the solution of Navier-Stokes equations in twodimensional domains. Lima [19] added to this code the simulation methodology for large scales of turbulence in two-dimensional flows. In order to discretize the domain, the quadrilateral nine nodes element was used. The main purpose of this work is the numerical implementation of a controlvolume-based on finite-element method for simulation of three-dimensional convection-diffusion problems using quadratic complete 27 nodes finite elements for domain discretization.


MATHEMATICAL FORMULATION Shown below, is the mathematical development for the equation involved in this work. The equation that characterizes the problem investigated is the transport equation, which may be generally written by using Cartesian tensor notation, such as qðq /Þ qðq ui /Þ q q/ þ ¼ C/ þ S/ qt qxi qxi qxi

ð1Þ

In Eq. (1), ui represents the velocity components along the coordinates axes xi, q is the specific mass (constant), C/ is a diffusion coefficient that depends on which variable / is being transported, and S/ are source terms that may encompass other terms, including differential ones, which are not explicitly written. An asterisk indicates dimensional variables. Variable / in Eq. (1) may represent any scalar variables, for example, the temperature T, turbulent kinetic energy, viscous or specific dissipation rate, or even the concentration of a contaminant in a certain medium or any component of Navier-Stokes equations. This work presents results with the variable / being handled as the temperature distribution and contaminant concentration in a certain medium. Dimensionless Form of Variables in Equations The specification of physical properties for the most diversified fluids is a difficult task in the numerical implementation of flow equations, so equations for dimensionless variables is generally preferred. When the solution of equations in dimensionless variables is chosen, it results in dimensionless parameters defined according to the geometrical, kinematical, and dynamic characteristics of each case considered. The dimensionless magnitudes for the case addressed herein may be defined as follows. Xi ¼

xi L

Ui ¼

ui u0

t¼

t u0 L

U¼

ð/ /0 Þ D/

q¼

q q0

C/ ¼

C/ C0

ð2Þ

where, L is the characteristic length of the domain and u0 is a characteristic flow velocity. Each magnitude presented in Eq. (2) refers, respectively, to space, velocity,

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O. A. NEVES ET AL. Table 1. Variables, properties and terms of Eq. (3) Name Heat Mass

U

CU

T T0 DT C C0 C¼ DC

m m þ t Pr Prt m m þ t Sc Sct

SU Heat generation Chemical reactions

and time dimensionless parameters for any scalar value of the density and diffusion coefficient, respectively. By using the dimensionless magnitudes, Eq. (1) may be rewritten as follows.


qðqUÞ qðUj qUÞ q CU qU þ ¼ þ SU qt qXj Re qXj qXj

ð3Þ

Some variables represented in Eq. (3) are shown in Table 1. The dimensionless parameters, Reynolds number, Re, Prandtl number, Pr, and Schmidt number, Sc, are defined in function of reference properties, as follows. Re ¼

q0 u0 L m0

Pr ¼

m0 cp0 k0

Sc ¼

m0 q0 D0

ð4Þ

Where Pe ¼ Re Pr for the heat transfer case (Table 1). After some algebraic handling and simplifications, Eq. (3) may be rewritten as qðUÞ qðUj UÞ q 1 qU þ ¼ þ SU qt qXj Pe qXj qXj

ð5Þ

DEVELOPING THE NUMERICAL MODEL This work develops a control-volume-based on finite-element method (CVFEM) by using the 27 nodes hexahedron as the reference element. Here, for each node in the finite-element mesh, there is a control volume around the node in which the unit value for the weight function is assigned. Out of the control volume, the weight function is null. Consequently, the control-volume-based on finite-element methods (CVFEM) involve the imposition of physical laws, such as energy conservation, over finite control volumes within the calculation domain. Thus, CVFEMs meet the global conservation principles. The application of the finite-element method with the formulation by control volumes, as described by reference [9], is made according to these basic steps: 1) domain discretization in finite elements; 2) further domain discretization in control volumes with each control associated to each node; 3) integration of equations in the control volumes; 4) specification of interpolation functions for the variables within the integrals and derivation of an algebraic system of equations; and 5) selection of one method to solve the resulting algebraic system.


735


Figure 1. A 27-node hexahedron.

A 27-node hexahedron was used as the element for the discretization of the three-dimensional domains of the equations involved. This is a complete quadratic finite-element shown in Figure 1. By only considering the corner nodal points, we have a linear finite element in both directions. Inside one finite element, there are 26 pieces of control volumes; here named sub-control volumes: eight associated to the corner nodes, six associated to the faces nodes, and 12 associated to the nodes at the middle of the edge sides. There is a complete control volume around the central node. Next, the discretization of equations is made. The integration of the governing partial differential equations by CVFEM is made in the sub-control sub-volumes within each element. Firstly, discretization of the transient term is made. For such, several schemes may be used: from an explicit to a fully-implicit scheme. On the one hand, the fully explicit scheme relies on limitations in terms of time increment due to solution stability reasons. On the other hand, the fully-implicit scheme enables setting the time increment according to the desired solution accuracy. In principle, there are no limitations in terms of time increment in this scheme which is unconditionally stable. Time discretization is defined by a parameter h, which will play the indicator role in the schemes; that is, that the parameter is to be used to indicate whether the scheme is explicit or implicit. Let Un as the field of temperatures (or concentration) defined in time tn nþ1 and t ¼ tn þ Dt. From Un and the specified boundary conditions, the field Unþ1 is calculated by the equation below: ðUÞnþ1 q þh qXj Dt

ðUj UÞ

nþ1

1 qðUÞnþ1 Pe qXj

!! ¼ hðSU Þnþ1 þ RnU

ð6aÞ

where RnU ¼

qðUj UÞn ðUÞn q 1 qðUÞn ð1 hÞ ðSU Þn qXj Pe qXj Dt qXj

ð6bÞ

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O. A. NEVES ET AL. Table 2. Values of parameter h, scheme indicator Schemes 0 1=2 2=3 1


h

Accuracy order

Explicit, conditionally stable Crank–Nicolson, stable Galerkin, stable Fully-implicit, stable

O(Dt) O(Dt)2 O(Dt)2 O(Dt)

Some values usually adopted for h are provided by reference [20] and defined in Table 2. When Galerkin’s finite-element method is used the spatial discretization of equations is made from their weak formulation. This weak form of equations is obtained by making the scalar product of the equation terms by weight functions, and integrating them by parts on the domain, in order to decreasing the secondorder operator degree. In the case of the control-volume-based on finite-element method, the weight function is made constant and unitary within each control volume. Thus, from the integration by parts of Eq. (6a), we have Z

! I ðUÞnþ1 1 qðUÞnþ1 nþ1 dV þ h ðUj UÞ nj dA Pe qXj Dt V A Z Z ¼h ðSU Þnþ1 dV þ ðRU Þn dV V

ð7aÞ

V

where Z V

RnU dV ¼

Z

ðUÞn dV ð1 hÞ V Dt

Z Z 1 qðUÞn ðUj UÞn nj dA þ ð1 hÞ ðSU Þn dV Pe qXj A V ð7bÞ

In Eqs. (7a) and (7b), nj represents the components of the normal vector outward to the areas. The normal vector can be defined as ~ k dA j þ n3~ ndA ¼ n1~i þ n2~

ð8aÞ

which, for three-dimensional cases results in ~ ndA ¼ dydz~i þ dxdz~ j þ dxdy~ k

ð8bÞ

In order to obtain the algebraic system of equations, Eq. (7a) is applied to each sub–control volume inside an element. The complete system of algebraic


737

equations is obtained by adding the contribution element-by-element. This procedure facilitates the obtainment of global matrices and does not affect the conservation principle, since when the contribution from each element for all nodes is considered, the contribution for complete control volumes is considered too. The interpolation functions used in this work may be found in reference [21]. Thus, by considering the equations for a control sub–control volume associated to a certain node a of an element, we have " NF Z X ðUÞnþ1 dV þ h Dt Va AKa K¼1

Z

Uj U

nþ1

# ! Z 1 qðUÞnþ1 ðSU Þnþ1 dV nj dA h Pe qXj Va

ðRUa Þn þ similar contributions from other elements for node a


þ contributions from boundaries (if applicable) ¼ 0:

ð9Þ

In Eq. (9), the expression for Rn/a is

RnUa

(

NF Z X n 1 qðUÞn ðUÞn dV ð1 hÞ ¼ Uj U nj dA Pe qXj Va Dt AKa K¼1 Z ðSU Þn dV Z

ð10Þ

Va

By substituting the interpolation functions [21] in Eq. (9), we obtain the system below in scalar form for one element. e M e e nþ1 þ hðC K Þ ðUe Þnþ1 Dt e nþ1 n M e e nþ1 ð1 hÞðC K Þ ¼ ðUe Þn þ h SUe þ ð1 hÞ SUe Dt

ð11Þ

Where Me is the mass matrix, Ce is the convective matrix, Ke is the diffusive matrix, and S/e is the source term vector. In Eq. (11), the elements of the matrices are defined as Mab ¼ n Cab ¼

Kab ¼ e SUa ¼

Z I I

Z

Nb dV

ð12aÞ

Nb Ujn nj dA

ð12bÞ

1 qNb nj dA Pe qXj

ð12cÞ

SUe dV

ð12dÞ

VSVCa

ASVCa

ASVCa

VSVCa

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O. A. NEVES ET AL.

Once defined the interpolation functions Na(n, g, 1), the matrices defined in Eqs. (12a–12d) may be calculated for obtaining the coefficients of the matrices in the elements. Before assembling the global matrix, matrix calculation is required in the elements. This work is made easier by mapping each element of the real domain in a reference element, named master element. This mapping is made in function of the global coordinates of the element nodes and the interpolation functions defined in local coordinates in the element. Global coordinates within the element are defined by Xie ¼

NNEL X

Nae ðn; g; 1ÞXia

ð13Þ

a¼1


where Xia, i ¼ 1, . . . , Ndim, represents the coordinates of node a and Na are interpolation functions for the hexahedron. An integral of a function in the real domain can be transformed in an integral in local coordinates in the master element as Z

f ðx; y; zÞdxdydz ¼ V

Z

f ðn; g; fÞdetðJÞdndgdf

ð14Þ

V

where det(J) is the determinant of Jacobian matrix for the transformation from global to local coordinates define by 2 qx ½J ¼

qn 6 qx 4 qg qx qf

qy qn qy qg qy qf

3

qz qn qz 7 qg 5 qz qf

ð15Þ

The derivatives of any function in relation to global coordinates may be calculated by derivatives in local coordinates, and can be found in reference [21]. The matrices in elements are calculated during the solving process. Part of the calculations required, such as the values of interpolation functions and their derivatives in Gaussian points in the volumes and boundaries of sub-control volumes, are performed once and stored in temporary files, which are read during the calculation process for the element matrices. This procedure has the purpose of reducing computational time to partially make up for the time spent by the frontal method, which carries out data reading from disk. For a computational system with sufficiently large processing memory, data could be stored in its own memory which could accelerate the calculation process. The solution method adopted was the frontal method, as described in reference [22]. This method has the advantage of not requiring full assembly of the global matrix at any time, whereby the larger matrix assembled is defined by a parameter that defines the front size. Hence, the system solution may be performed in computers with relatively small memories and medium disk storage capacity.


739

NUMERICAL APPLICATIONS—THREE-DIMENSIONAL CONVECTIVE-DIFFUSIVE CASES This work will present a numeric study of the transient three-dimensional convection-diffusion governed by the following equation.


qU qðUUÞ qðV UÞ qðW UÞ þ þ þ qt qX qY qY q 1 qU q 1 qU q 1 qU ¼ þ þ þ SU qX Pe qX qY Pe qY qZ Pe qZ

ð16Þ

Two cases will be analyzed for such study. The first case will be the convectiondiffusion in a cubic cavity that posses results in open literature, thus enabling comparison and validation of the computational code. To do that, we will use the results presented in the work of Ledain Muir and Baliga [23]. In the second case, a single-channel convection-diffusion will be studied which will adopt a velocity profile for laminar flow, described in Shah and London [24]. The problem of cooling a hot fluid along the channel is considered. In all cases, prescribed velocity profiles have been adopted from the literature; no computation of the Navier-Stokes equations has been done. No special scheme has been applied for discretization of the convective terms. Convection–Diffusion in a Cubic Cavity (U: Temperature) The domains for this case are illustrated in Figure 2 and were presented by Ledain Muir and Baliga [23]. In the three cases, the cube with unit edges was divided into two equal parts by the ABCD plan. The boundary conditions for calculating the temperature U were defined as U ¼ 1 on one side of the ABCD plan and on the other side U ¼ 0, on the points where this plan intercepts the boundary of the domain, U ¼ 0.5. The discretization used a mesh with 1,000 elements resulting in 9,261 nodal temperatures.

Figure 2. Geometry and boundary conditions: (a) case 1, (b) case 2, and (c) case 3.

740

O. A. NEVES ET AL.

The solution of Navier-Stokes equations in the three-dimensional case is still being developed. Thus, velocity profiles were prescribed to solve the Eq. (16). According to the work by Ledain Muir and Baliga [23], the velocity profiles in the three cases considered are Case 1 1 u ¼ pffiffiffi 2

1 v ¼ pffiffiffi 2

w¼0

ð17Þ

Case 2 1 u ¼ pffiffiffi 3

1 v ¼ pffiffiffi 3

1 w ¼ pffiffiffi 3

ð18Þ

1 u ¼ pffiffiffi 3

1 v ¼ pffiffiffi 3

1 w ¼ pffiffiffi 3

ð19Þ


Case 3

The temperature profile for fixed x, y positions, x ¼ y ¼ 0.5 in function of z can be observed in Figures 3–5 in the velocities of cases 1–3, respectively. As already expected for convective-diffusive processes, for problems with low Peclet number the diffusion process is dominant, and the expected behavior of the temperature is correctly predicted. However, with the increasing of the Peclet number, the problem becomes convective-dominating presenting oscillations, especially for coarse meshes. Figure 6 presents a comparison of the results from this study with the results from reference [23], and the exact solution for the problem in the case of velocity in case 3. There is good agreement between the results.

Figure 3. Temperature profile, case 1.


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Convection–Diffusion in a Channel (U: Temperature) A channel of 40 10 10, that is 0 x 40 and 0 y, z 10 has been adopted as domain. For discretization, 4,000 elements have been used, which have resulted in 35,721 nodal unknown temperatures. A velocity profile for laminar flow in rectangular ducts as described in reference [24] was adopted to represent the velocity field. This profile presented below for the proposed channel has the following form. y2;2 z 2;2

u ¼ 1 1 ð20Þ umax 5 5



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O. A. NEVES ET AL.

Figure 6. Temperature profile for Pe ¼ 10000: (a) case 1, (b) case 2, and (c) case 3.

To the boundary condition at input, plan x ¼ 0, is U ¼ 1, and in the other plans except for plan x ¼ 40 which is the output, a boundary condition for scalar U ¼ 0 has been imposed. Figure 7 illustrates the geometry for this case. The results to be presented refer to the temperature calculation. Figure 8a shows the influence of the Peclet number on the temperature. At the coordinates considered,

Figure 7. Geometry of the channel.


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Figure 8. Temperature profile for (a) different Pe and (b) with Pe ¼ 52 on different plans of constant x.

there are small differences near the central region. Figure 8b presents the temperature profile development on plans x ¼ 5, 10, 15, 20, 25, and 30 and y ¼ 5 in function of z. As expected, we observe that as x increases, the temperature decreases due to the boundary conditions imposed that represent a cooling fluid problem. To illustrate a two-dimensional view of the problem, the plans x ¼ 10 and x ¼ 30 were selected for Pe ¼ 52 (Figures 9a and 9b). We may observe in Figures 9a and 9b the occurrence of cooling along the x axis. Convection–Diffusion in a Channel (U: Concentration) In this application, by using the same domain of Figure 7 as the application, results were obtained for the dispersion of a contaminant with polluting source set on an element located on the soil, close to plan x ¼ 10, for a Schmidt number equal to 0.2. The polluting source, with value C ¼ 1, was set at point (x, y, z) ¼ (10, 5, 1).

Figure 9. Temperature profile on (a) plan x ¼ 10, left and (b) x ¼ 30, right.


744

O. A. NEVES ET AL.

Figure 10. Concentration lines along the channel.

Figure 10 shows the concentration behavior along the channel for several values of x, fixed y ¼ 5, and 0 z 10. The concentration leap results from the matching with the location where the polluting point was set. We observe in Figure 10 the concentration is closer to value C ¼ 1 (polluting source) as x value is closer to x ¼ 10. Figures 11–13 show the concentration field at some plans of constant x coordinate.

Figure 11. Concentration field on plan x ¼ 15.



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As shown in Figures 11–13, the more distant the plan is from the contamination point, the lower the values presented in concentration fields. Convection–Diffusion in a Channel (U: Temperature) The last case investigated in this work refers to the situation where the origin of the Cartesian coordinates is, as shown in Figure 14. Thus, the channel was divided


746

O. A. NEVES ET AL.


Figure 14. Geometry and boundary conditions in the channel.

Figure 15. Temperature profile for different Pe at (a) x ¼ 5, y ¼ 0.5, and (b) x ¼ 15, y ¼ 0.5.

Figure 16. Temperature profile for Pe ¼ 52.


747

into four identical channels and the temperature profile was analyzed in only one quart of the channel due to the symmetry condition imposed on the wall. The dimensions of the channel adopted for analysis are 40 1 2 and 4,000 elements were used in the discretization, which implies in 35,301 nodal unknowns temperatures. Figure 14 illustrates the geometry and boundary conditions in a cooling problem. The velocity profile chosen to represent the velocity field in this case in according to with [24] has the following form. u umax

¼ 1y

2;05

z3;02

: 1 2

ð21Þ


The temperature profile for Peclet numbers 52, 72, and 100 at y ¼ 0.5, x ¼ 5, and 15 is shown in Figures 15a and 15b, respectively. Figure 16 shows the profile development in function of z axis for the Peclet number equal to 52 on some plans in x. We are able to observe the behavior and trends expected in this kind of process is simulated. CONCLUSION An implementation of a CVFEM for three-dimensional convection–diffusion problems has been done in this work. The reference finite element used for spatial discretization was the quadratic 27 nodes hexahedron element. The first application presented is a traditional case of fluid mechanics for a cubic cavity. This application presented good results when compared to the exact solution, especially when the problem is not highly convective. For a dominating convection situation, the numerical oscillations begin to appear. These oscillations may be a consequence of not using any special scheme for discretization of the convective terms, or because of the relatively gross meshes used. All simulations were done on small computers and the use of much refined meshes was not possible. For convection–diffusion cases in channels also with Cartesian geometries, the computational code for the controlvolume-based on finite-element methods presented good results in qualitative terms once it was capable of simulating the temperature profiles for a cooling problem along the flow axis and evidenced that, for contaminant concentration cases, the more distant from the contamination point the less expressive will be the pollutant concentration field. In future works, the implementation for the complete Navier-Stokes equations should appear. REFERENCES 1. C. T. Shaw, Using Computational Fluid Dynamics, Prentice Hall, New York, 1992. 2. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, 1980. 3. J. B. Campos-Silva, Simulaç aõ Nume´rica de Escoamentos de Fluidos pelo Me´todo de Elementos Finitos Baseado em Volumes de Controle, Doctorate Thesis in Mechanical Engineering, UNICAMP—Faculdade de Engenharia Mecaˆnica—Departamento de Engenharia Te´rmica e de Fluidos, Campinas-SP, Brasil, 1998. (In Portuguese.) 4. G. Dhatt and G. Touzot, The Finite-Element Method Displayed, John Wiley & Sons, Chichester, 1984.


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Applications Numerical Simulation of Convection

Applications Numerical Simulation of Convection

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