CHAPTER 3 Multi-domain DRM boundary element method for the … · 2014-05-10 · CHAPTER 3 Multi-domain DRM boundary element method for the numerical simulation of non-isothermal

CHAPTER 3

Multi-domain DRM boundary element methodfor the numerical simulation of non-isothermalNewtonian and non-Newtonian flow problems

H. Power1 & W.F. Florez21The University of Nottingham, School of Mechanical, Materials andManufacturing Engineering, Nottingham, UK.2Energy and Thermodynamics Research Group, UPB University,Medellin, Colombia.

Abstract

The multi-domain dual reciprocity method has been effectively applied to the solu-tion of two-dimensional non-isothermal Newtonian and non-Newtonian flow prob-lems, where the momentum and energy equations govern the motion of a viscousfluid. In the proposed boundary integral method the domain integrals are trans-formed into equivalent boundary integrals by the dual reciprocity approach appliedin a subdomain basis. On each subregion or domain element the integral repre-sentation formulae for the velocity and temperature are applied and discretizedusing linear continuous boundary elements, and the equations from adjacent subre-gions are linked by additional continuity conditions. Some examples showing theaccuracy, efficiency and flexibility of the proposed method are presented.

1 Introduction

The main purpose of this work is to present the use of the multi-domain dual reci-procity method (MD-DRM) [1, 2] for the numerical simulation of non-isothermalNewtonian and non-Newtonian flow problems with viscous dissipation effects. TheMD-DRM is a recently developed technique that allows an efficient and accurateboundary element numerical solution of highly nonlinear and convective problems.In this work, we will show how the MD-DRM technique can be used to solve this

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70 Domain Decomposition Techniques for Boundary Elements

type of problems, showing the efficiency and versatility of the proposed numericalapproach.

The application of the boundary element method (BEM) to Newtonian or non-Newtonian nonlinear problems requires finding a fundamental solution of the sys-tem of governing equations. However, since such a fundamental solution for ageneral model is not possible to be known, it is necessary to put the nonlinear termsinto a pseudo-body force leading to domain integrals that can be evaluated by usingthe cell integration approach (cell–BEM) [3]. Although this method is effective andgeneral, it makes the BEM lose its boundary-only nature resulting in a numericalscheme several orders of magnitude more time consuming than other domain tech-niques as the finite differences method (FDM) and finite element method (FEM).The computational cost of the cell integration approach mainly depends on the factthat the solution at each surface or internal point must involve the evaluation of thecomplete surface integrals yielding in this way to a fully populated matrix system.

A handful of methods have been developed to take the domain integrals to theboundary. Among them, the dual reciprocity method (DRM) [4] and the particularsolution method (PIT) [5] are the most popular ones. These two approaches havebeen particularly attractive in recent years because of the advances in multidimen-sional interpolation with radial basis functions (RBF) used in these techniques.Both the DRM and PIT approximate the nonlinear and non-homogeneous terms ofa partial differential equation as a series of vector-valued interpolation functions,which are defined in terms of a set of surface and internal nodes. These interpolationfunctions lead to a particular solution for the problem which can be used togetherwith Green’s identities to convert the domain integrals into boundary integrals.As in the cell integration approach, in these two boundary-only approaches theevaluation of the functions at each node requires the integration of the completesurface integrals yielding again a fully populated matrix system.

A major problem encountered with the DRM and PIT is that the resultingalgebraic system consists of a series of matrix multiplications of dense matrices.Although the computing time required by the DRM and PIT is lower than that withthe cell integration scheme, it is still very high compared to domain approaches.Besides, in complex problems both the DRM and PIT have been limited to a smalldegree of nonlinearity. From these limitations, it appears that this type of boundary-only formulations in their original single domain form cannot be extended to solvehighly nonlinear problems.

When dealing with the BEM solution of large problems it is usual to use themethod of domain decomposition, in which the original domain is divided intosubregions, and in each of them the full integral representation formula is applied.At the interfaces between the adjacent subregions, the corresponding continuityconditions are enforced. While the matrices which arise in the single domain DRMare fully populated, the subregion formulation leads to block-banded matrix systemswith one block for each subregion and overlaps between blocks when sub-regionshave a common interface.

The implementation of the subregion BEM formulation in the limiting case of avery large number of subregions, including cell integration at each subregion was



Multi-Domain DRM Boundary Element Method 71

called by Taigbenu and collaborators as the Green element method (GEM) [6]. Inthis method, the resultant coefficient matrix is as sparse as that of the finite elementmethod and, hence, its solution is as efficient as in the domain approach.

Recently, Popov and Power [1] found that the DRM approximation of the internalpotential of a nonlinear convection diffusion problem can be substantially improvedby using a domain decomposition scheme. Popov and Power’s idea of using domaindecomposition to improve the accuracy of the DRM approach was inspired by thework of Kansa and Carlson [7] on data approximation with RBFs. Kansa andCarlson observed that the best approximation is obtained when the original domainis split into matching subdomains. In a similar way, Popov and Power [1] noticedthat the DRM approximation for nonlinear problems can also be improved whenthe original domain is divided into smaller subdomains. At each subregion theBEM formulation of the problem is separately applied. The corresponding domainintegral resulting from the BEM formulation is transformed to surface integralsalong the contour of the subregions by the DRM approach. The desired accuracy ofthe scheme is achieved by refining the size of the subdomains. For a large numberof subdomains this technique was called a MD-DRM, and the discrete form of theintegral equation formulation of the problem is given by a banded matrix system.Recently the MD-DRM has been successfully applied by Florez and Power [8]to the solution of isothermal non-Newtonian flow problems and to the Navier–Stokes equations for incompressible Newtonian fluids at moderately high Reynoldsnumbers [9]. In the present work it will be shown how the MD-DRM can be extendedto more general non-isothermal problems including viscous dissipation terms.

The problem of multiplication of several large fully populated matrices encoun-tered in the DRM formulation in a single domain is reduced, in the present domaindecomposition case, to the multiplication of very small matrices at each subdomain.Besides, if the subdomains are of similar shape all the evaluations and multiplica-tions can be performed only in one subdomain, and the results for the others can befound by scaling due to the geometric characteristic of the DRM functions. Becauseof the nature of the integral equation formulation at each subdomain, the solutionobtained with MD-DRM is a kind of Hermitian solution. In the present case ofnon-isothermal flow, the values of the velocity and shear stress fields as well as thetemperature and its directional derivatives at every point defining the subdomainsare found directly from the solution of the resulting matrix system.

Although the solution of a fully populated system of equations is computationallyvery expensive, this type of solution for a sparse system can be carried out veryefficiently by not taking into account the multiplications of the zero terms. In thismanner the solution of the MD-DRM system could be obtained in a fraction of thesolution time of an equivalent fully populated system resulting from any of the otherstandard BEM schemes. Besides, the required storage of the MD-DRM approach issubstantially smaller than the one required by other BEM schemes. For the sparsesystem obtained with the MD-DRM approach, both the efficiency and the storagerequirement are functions of the bandwidth of the matrix system. It is importantto point out that the solution of any BEM approach could also be substantiallyimproved with the use of indirect solvers which were not considered in this work.




In the following sections, we will introduce the MD-DRM solution of the coupledsystem of Navier–Stokes equations and the energy equation. Besides the efficiencyand accuracy of the proposed numerical method, the MD-DRM has the advantageof the robust RBF interpolation used in the DRM. Due to their smoothness ornoise minimization character, the radial functions guarantee high accuracy on theevaluation of the gradient of the field variables [10]. This property of the DRMinterpolation permits a precise evaluation of the velocity and temperature derivativesdirectly from the solution of the matrix system without any additional burden.

2 Thermal convection Newtonian flow problems

In natural convection phenomena the flow pattern and the velocity field of a viscousfluid is established by buoyancy forces caused by temperature differences withinthe fluid. For most incompressible Newtonian fluids the viscous dissipation and thetemperature dependence of the viscosity can be neglected in the energy equation. Incontrast, the density variation represents an important term that acts as a body forcein the momentum equations and causes the fluid motion. Buoyancy is due to thecombined presence of a fluid density gradient and a body force that is proportionalto the density. The body force is usually gravitational, although it might be anyother force such as centrifugal or a Coriolis force. There are several ways in whicha mass density gradient may appear in a fluid but in the most common situation itis due to a temperature gradient. Although in the present work the fluid is goingto be assumed incompressible, the effect of variable density in the buoyancy forcewill be accounted for using the Boussinesq approximation.

Free convection flows may be classified according to whether the flow is boundedby a surface. In the absence of bounding surfaces, free boundary flows may occurin the form of plumes or buoyant jets [11]. In this work we focus on free convectionflows of incompressible fluids bounded by a surface. In this type of problems athermal boundary layer usually appears on the heated walls and in those zonesthe high convection terms in the momentum and energy equations dominate thesolution.

In the past, several BEM formulations based on cell integration have been suc-cessfully applied to the solution of the thermally driven cavity flow problem. Onishiet al. [12] proposed a cell–BEM formulation for natural convection problemsin terms of the stream function, vorticity and temperature as the main variables.On the other hand Skerget et al. [13] used a velocity, vorticity and temperatureformulation together with domain partition techniques and discontinuous bound-ary elements for the solution of both Newtonian and non-Newtonian flow problems.The penalty function formulation is another method that can be used to solve fluidflow problems and it was presented separately by Kuroki [14], Brebbia [15] andKitawa [16]. In this formulation a penalty parameter is used to obtain a separateequation for the pressure and so the Navier–Stokes equations are reduced to theNavier equations of elastostatics with a body force including the convective andbuoyancy terms.




The aim of this section is to show an efficient use of the MD-DRM technique forthe solution of the coupled momentum and energy equations. The domain integralsthat appear on each subdomain can be converted into a series of boundary integralsby dual reciprocity.

2.1 Governing equations

The governing equations of the fluid flow (continuity, momentum and energy)assuming the Boussinesq approximation (density variations are relevant only in thebuoyancy terms of the momentum equations), Newtonian fluid behaviour, negli-gible influence of pressure on temperature and constant fluid properties, may bewritten in tensor notation as:

∂ui

∂xi= 0 (1)

− ∂p

∂xi+ µ ∂2ui

∂xj∂xj= ρuj

∂ui

∂xj− ρgiβ(T − Tc) for all x ∈ � (2)

k∂2T

∂xj∂xj= ρCpuj

∂T

∂xj(3)

where ui is the flow velocity, ρ the fluid density, p the pressure, µ the viscosity,and � is a bounded two-dimensional domain. In the energy equation (eqn (3))T is the fluid temperature at any point, k is the thermal conductivity and Cp thethermal capacity. The lifting force ρgiβ(T − Tc) in eqn (2) represents the effect oftemperature change on the density as a buoyancy term, where Tc is the referencetemperature, i.e. the temperature at the cold wall in a convective flow problem, βis the coefficient of thermal expansion and gi is the magnitude of gravity acting inthe i direction.

Let us now consider the dimensionless parameters that govern free convectiveflow and heat transfer. These parameters can be obtained by non-dimensionalizingthe governing equations. Defining,

x∗i = xi

Lu∗

i = uiL

αθ = T − Tc

Th − Tcp∗ = pL2

µα(4)

where α is the thermal diffusivity and L and Th are a reference length and thereference temperature of a hot wall, respectively. With these new variables thecontinuity, momentum and energy equations (eqns (1)–(3)) reduce to,

∂u∗i

∂x∗i

= 0 (5)

−∂p∗

∂x∗i

+ ∂2u∗i

∂x∗j ∂x∗

j= 1

Pru∗

j∂u∗

i

∂x∗j

− Raθ for all x ∈ � (6)

∂θ

∂x∗j ∂x∗

j= u∗

j∂θ

∂x∗j

(7)




and the following dimensionless numbers have been defined:

Gr = βgi(Th − Tc)L3

ν(8)

Pr = ν

α(9)

Ra = Gr Pr (10)

In the above equations, Gr is the Grashof number, Pr is the Prandtl number, ν isthe kinematic viscosity and Ra is the Raleigh number. For simplicity, from now on,we adopt the notation u∗ = u, p∗ = p and x∗

j = xj. The Grashof number plays thesame role in free convection that the Reynolds number plays in an isothermal fluid,and the former indicates the ratio of the buoyancy force to the viscous force actingon the fluid.

In general, the flow field may satisfy mixed boundary condition of the form

ui(ξ ) = Ui(ξ ) for all ξ ∈ SD (11)

and

ti(ξ ) = σij(u, p)nj(ξ ) for all ξ ∈ SN (12)

where S = SD + SN is the boundary surface with outward unit vector n, t is thesurface force or traction and σij is the stress tensor defined as:

σij = −pδij + µ(∂ui

∂xj+ ∂uj

∂xi

)(13)

And there is also a set of boundary conditions for the energy equations given by,

T (ξ ) = Te for all ξ ∈ SA (14)

and

q = −k∂T

∂n= qe for all ξ ∈ SB (15)

where S = SA + SB is the boundary surface. Note that eqns (14) and (15) correspondto specified temperature and heat flux along different parts of the boundary S.

2.2 Integral representation formulae

The integral representation formulae for the eqns (1) and (2) at a point x ∈ � ∪ Sare given by Ladyzhenskaya [17]:

ckj(x)uk(x) =∫

SKkj(x, y)uj(y)dSy −

∫S

Uki (x, y)tj(y)dSy +

∫�

Uki (x, y)gi(y)d�

(16)




here g is the non-homogeneous term or pseudo-body force equal to the convectiveterm, i.e.

gi = ρuj∂ui

∂xj− ρgiβ(T − Tc) (17)

The coefficients cki have values between δki and 0, being equal to 12δki for smooth

boundaries and cki = δki for points inside the domain �. The Green’s function Uki

under the second integral in eqn (16) is the fundamental solution of the Stokes’equation known as Stokeslet, with a corresponding pressure qk :

Uki (x, y) = − 1

4π

(ln

(1

r

)δij + (xi − yi)(xk − yk)

r2

)(18)

qk(x, y) = − 1

2π

(xk − yk)

r2 (19)

and Kij represents the traction fundamental solution given by

Kij(x, y) = σ ′ik(U j

i (x, y), q j(x, y))nk(y) (20)

where

σ ′ik(U j

i (x, y), q j(x, y)) = δlkq j(x, y) + µ(∂U j

i

∂xk+ ∂U j

k

∂xi

)(21)

and thus

Kij(x, y) = − 1

π

(xi − yi)(xj − yj)(xk − yk)

r4 nk(y) (22)

The parabolic partial differential equation (3), can be seen as a Poisson equationwhose integral representation is obtained by applying Green’s second theorem fora scalar function [18]. However, due to the presence of the convective term in eqn(3), a domain integral is present in the integral representation and this latter integralcan be transformed to the boundary using the DRM technique [4]. The integralrepresentation of the energy transport equation at each domain element in whichthe domain will be divided, can be expressed as follows:

cT (x) =∫

SH(x, y)T (y)dSy −

∫S

G(x, y)q(y)dSy +∫�

G(x, y)h(y)d�y (23)

where

q(x) = ∂T

∂n(24)

h(x) = uj∂T

∂xj(25)




and G and H are the fundamental solution and its normal derivative along theintegration path, respectively, i.e.

G(x, y) = 1

2πlog r(x, y) (26)

and

H(x, y) = 1

2π

∂ log r(x, y)

∂ny(27)

where x and y are the position vectors of the source and field points, respec-tively [3].

2.3 Approximation of the domain integrals

As was explained in our previous articles [9, 19], the non-homogeneous terms thatappear within the domain integrals in eqns (16) and (23) can be approximatedusing a series of particular solutions and interpolation functions. To express thedomain integral in eqn (16) in terms of equivalent boundary integrals, the DRMapproximation is introduced. The basic idea is to expand the g(y) term using radialinterpolation functions at each subregion, i.e.

gi(x) =N+L+A∑

m=1

f m(x)αml δil (28)

The coefficients αml are unknowns to be determined by collocation on a set of N

nodes on the boundary and L internal nodes. It will be considered here that there areA = 3 augmentation global functions from the set {1, x1, x2}. It should be pointedout that the functions x1 and x2 alone give results that are not translatory symmetric.However, in the present work, the mentioned augmenting functions are calculatedin a local system of coordinates whose origin is at the centroid of each subdomain.Thus, the global augmenting functions are different for each cell and scaled to avoidany numerical problems.

An augmented spline consists of the RBF plus a series of additional global func-tions [20]. The RBF used in this work is the thin-plate spline:

f m(x) = f (r(x, ym)) = r2 log r m = 1, . . . , N + L (29)

where r = r(x, ym) is the Euclidean distance between the field point x and the collo-cation point ym, other RBFs are discussed in the literature [20], such as Gaussian andmultiquadratic functions. However, the thin-plate spline has features of smoothnessand convergence that make it the recommended choice.

Equation (28) when applied to the N + L collocation nodes will generate2(N + L) linear equations with 2(N + L + A) unknowns and, therefore, 2A addi-tional conditions are necessary which basically guarantee the maximum smoothness




of the interpolant [20]. These additional relationships are:

N+L∑j=1

αjl δil =

N+L∑j=1

x j1α

jl δil =

N+L∑j=1

x j2α

jl δil = 0 (30)

where xj represents the jth collocation node.Similarly, for the energy eqn (23) the convective term can be approximated as:

h(x) = ρCpuj∂T

∂xj=

N+L+A∑m=1

f m(x)βm (31)

with the following additional equations for the global interpolation functions,

N+L∑j=1

β j =N+L∑j=1

xj1β

j =N+L∑j=1

xj2β

j = 0 (32)

After defining the auxiliary velocity field (Ulmi (x), plm(x)) [21] solution of the

following equations:

µ∂2Ulm

i (x)

∂xj∂xj− ∂ plm(x)

∂xi= f m(x)δil (33)

∂Ulmi

∂xi= 0 (34)

and the auxiliary thermal field T solution of the following Poisson equation,

k∂2Tm(x)

∂xj∂xj= f m(x) (35)

we can apply Green’s identities to both the auxiliary velocity field and the auxiliarythermal field to obtain integral representation formulae in terms of only boundaryintegrals [4, 9, 19, 21], thus

ckj(x)uk(x) −∫

SKkj(x, y)uj(y)dSy +

∫S

Uki (x, y)tj(y)dSy

=N+L+A∑

m=1

αml

{ckj(x)Ulm

k (x) −∫

SKkj(x, y)Ulm

j (y)dSy +∫

SUk

i (x, y)tlmj (y)dSy

}(36)

and likewise for the energy equation,

cT (x) −∫

SH(x, y)T (y)dSy +

∫S

G(x, y)q(y)dSy

=N+L+A∑

m=1

βm{

cTm(x) −∫

SH(x, y)Tm(y)dSy +

∫S

G(x, y)qm(y)dSy

}(37)

where qm = ∂Tm/∂n.




2.4 Particular solution

The particular solution for the flow field (Ulmi (x), plm(x)) defined by the eqns (33)–

(34) have been presented in earlier papers by Mingo [22] and Florez [19], and itcan be obtained using the approach suggested by Power and Partridge [21]. In thischapter and based on our previous successful implementation of the MD-DRMtechnique using augmented thin-plate splines for the solution of the Navier–Stokesequations, we focus on the particular solution of the auxiliary Poisson equation, eqn(35). With the convective term interpolated using the functions f m(x) as in eqn (31),the particular solution Tm(x) in eqn (35) can be found using cylindrical coordinatesas suggested in [4] for the RBF, and the method of undetermined coefficients [20] forthe polynomial interpolation functions. The results are summarized as follows [23]:

Tm = r4 log r

16− r4

32for f m(x) = r2 log r (38)

where r = ‖x − ym‖ is the Euclidean distance between the field node x and thecollocation node ym. On the other hand, the particular solutions corresponding tothe global functions f m(x) = 1, x1, x2 are:

Tm = x21 + x2

2

4(39)

Tm = x31 + x1x2

2

8(40)

Tm = x32 + x2x2

1

8(41)

respectively. The corresponding normal derivatives are given by the followingequations,

qm = ∂Tm

∂n= r2xini log r

4− r2xini

16for f m(x) = r2 log r (42)

while for the global functions f m(x) = 1, x1, x2 the corresponding fluxes aregiven by,

qm = x1δ1i + x2δ2i

2ni for f m(x) = 1 (43)

qm = (3x21 + x2

2)δ1i + 2x2x1δ2i

8ni for f m(x) = x1 (44)

qm = (3x22 + x2

1)δ2i + 2x1x2δ1i

8ni for f m(x) = x2 (45)

where δij is the Kronecker delta.Having obtained the auxiliary non-homogeneous thermal field, we are now in a

position to handle the convective term of the energy equation by the DRM approach.




Taking into account that in eqn (31) the velocity uj is considered as a knownvariable at each iteration of the coupled problem, we only need to approximate thetemperature gradient ∂T/∂xj. Therefore, by expanding the temperature in terms ofthe interpolation functions f m(x), with unknown coefficients γm, we obtain

T =N+L+A∑

m=1

f m(x)γm (46)

or in matrix notation

T = Fγ (47)

where the components of vector T are the temperature values at each node, thecolumns of matrix F contain the interpolation functions and vector γ contains theunknown coefficients.

From eqn (47) it is easy to establish a mechanism that can relate the nodal valuesof T with the nodal values of its derivatives, ∂T/∂xj by taking the derivative of eqn(46) with respect to xj, thus

∂T

∂xj=

N+L+A∑m=1

∂f m(x)

∂xjγm (48)

Substitution of eqn (47) into eqn (48) yields,

∂T∂xj

= ∂F∂xj

F−1T (49)

Now we can express the unknown coefficients βm in the approximation of theconvective term that appears in eqn (31), as follows:

β = ρCpF−1uj∂F∂xj

F−1T (50)

where vector β contains the unknown coefficients βm and uj is a diagonal matrixwith nodal values of the velocity.

The non-homogeneous term of the momentum equations given in eqn (17)requires not only the values of the temperatures but also the calculation of thevelocity derivatives at the collocation nodes. In this work we have used a velocityinterpolation scheme that satisfies the mass conservation principle and that waspresented in [9]. In the mass conservative interpolation scheme the velocity deriva-tives are approximated in terms of the components of the velocity field, i.e. u1, u2in such a way that the mass balance (1) is satisfied in the domain. The main resultscan be summarized in the following equations (for more details see [9]):

g = Fα (51)




which is just eqn (28) written in matrix form. Likewise, the following approximationfor the velocity components holds,

u = Fλ (52)

where u contains the velocities at every node, matrix F contains the interpolationfunctions from the mass conservative interpolation method and λ is the vector ofunknown coefficients. Taking the derivatives on both sides of the latter equation weobtain,

∂u∂xj

= ∂F∂xj

F−1u (53)

With all the above approximations it is possible to discretize the integral equa-tions eqns (36) and (37) and obtain a system of equations for the velocities andtemperatures.

2.5 Discretization and numerical solution

Boundary integral equations (36) and (37) were discretized using linear isopara-metric linear boundary elements [18], hence, the momentum and energy integralequations can be rewritten as:

ciui −n∑

k=1

Kikuk +n∑

k=1

Uiktk =N+L+A∑

j=1

αj

{cjUj −

n∑k=1

KikUkj +n∑

k=1

Uik tkj

}(54)

and

ciTi −n∑

k=1

HikTk +n∑

k=1

Gikqk =N+L+A∑

j=1

β j

{cjTj −

n∑k=1

HikTkj +n∑

k=1

Gikqkj

}(55)

respectively, where Kik , Uik , Hik and Gik are the standard influence matrices result-ing from the integration over the boundary elements, the index i represents thecollocation nodes, k the nodes at the integration elements and j the DRM colloca-tion points.

For the numerical solution of the integral equations, we propose a multi-domainapproach that divides the entire domain into smaller subdomains or domain elementswhere each of them is enclosed by four boundary elements. On each subdomainthe eqns (54) and (55) hold. All the surface integrals are evaluated using Gauss–Legendre quadratures; however, special attention should be paid to the weaklysingular integrals whose kernel is the Stokeslet Uk

i (x, y) when the source pointtends to the field point x → y. This singularity can be dealt with using a spe-cial transformation of coordinates [24]. Besides the discretised integral equations,there are also additional continuity conditions at the interface between subdomains.




On the common interface �j that connects the subdomains i and i + 1 the velocitymatching conditions are of the form

ui�j

= ui+1�j

(56)

and the dynamic equilibrium or traction matching conditions are:

ti�j

= −ti+1�j

(57)

A similar procedure is applied to the integral equation for the temperature field;together with the temperature, Ti

�j= Ti+1

�j, and flux qi

�j= qi+1

�j, matching condi-

tions at the interface between the subregions.The unknown coefficients vectors αj and β j are determined from eqns (28), (31),

(50) and (51) by collocation on the boundary nodes of each subdomain. With theapproximations explained in Section 2.4, the vectorαj can be found in terms of nodalvelocities only, therefore, eqn (54) represents a nonlinear system for the tractionsand velocities. The resulting global nonlinear system of momentum equations forthe entire domain is solved using the Newton–Raphson scheme combined witha line-search algorithm intended to reduce the residual at each iteration [25]. Onthe other hand, we used simple or direct iteration to cope with the coupling ofthe momentum and energy equations through the buoyancy term. In this way thevelocity in the convective term from the energy equation is considered as a knownvariable from the previous solution of the momentum equations obtained by theNewton–Raphson method. Likewise, during each iteration of the iterative solutionfor the momentum equations, the non-homogeneous buoyancy force is consideredas a known variable. These forces are determined from the previous temperaturesolution by direct iteration. For more details about the numerical algorithm see [26].

The use of linear boundary elements in the context of the multi-domain tech-nique produces overdetermined systems of equations. Such systems can be solvedin a least squares sense either directly or by the normal equations. The direct solu-tion of the overdetermined set of algebraic equations is the simplest approach forfinding a least squares solution by methods such as the QR factorization or thesingular value decomposition [27]. Although these methods are robust for the caseof dense matrices and even ill-posed problems, they both require high amounts ofcomputer memory and time. Since in the multi-domain case the coefficient matrixis block sparse, an iterative solution such as that of Paige and Saunders [28] is atleast in theory, much faster at the same solution accuracy. However, the velocity–traction formulation of the momentum equations yields extremely ill-conditioned,non-symmetric and non positive definite matrices that make the iterative solverslose their advantages and speed unless the appropriate preconditioner is available.To avoid the numerical problems that may arise during the iterative solution ofoverdetermined systems of equations in the present investigation, a direct solverhas been applied to the normal equations. It is also worth mentioning that thenumerical ill-conditioning problems that exist for the momentum equations are notpresent in the solution of the scalar energy equations and so an iterative solver canbe used for the latter case.




2.6 Numerical results

To test the developed numerical method, the problem of natural convection flowin a square cavity is considered. Figure 1 shows the geometry of the problem andboundary conditions. The left wall is maintained hot at a temperature Th = 310 K,the right wall is at a colder temperature Tc = 300 K, the four walls are stationary,and the top and bottom boundaries are adiabatic. As the fluid next to the left wallgets warmer it ascends due to its lower density, while the fluid next to the cold wallmoves downwards. Both flows have to overcome the resistance due to viscosityand gravity, and as a consequence a big recirculation zone appears at the cavitycentre.

The natural convection within the cavity depends on two parameters: The Prandtlnumber Pr, and the Raleigh number Ra. The temperature difference between theheated walls is considered small enough to adopt the Boussinesq approximationwhich assumes that the fluid properties β, µ and Pr are constant and evaluated atthe average temperature (Tc + Th)/2. Although the density was assumed constant,the effects of its variations are included in the buoyancy term in the momentumequations.

Two different values of the Raleigh numbers Ra = 103 and Ra = 104 were con-sidered in the simulation. For each of the cases studied, two different mesh densitieswere used 10 × 10 and 20 × 20 divisions. Both cases, uniform and non-uniform

��

��

X

Y

q=0,U1=0,U2=0

q=0,U1=0,U2=0

θ=1,U1=0,U2=0 θ=0,U1=0,U2=0

g

L

L

Figure 1: Schematic of the thermally driven cavity problem.




grids, were studied in an attempt to capture the flow complexities near the walls.The distribution of nodes for non-uniform meshes is given by the expression [29]:

xi

L= i

imax− 1

2πsin

(2π

i

imax

)(58)

Even with the coarse meshes used in this work, the results obtained by thepresent MD-DRM technique are in excellent agreement with previous solutions[14,15,23,30] calculated with much denser meshes and the use of cell integration.The maximum and minimum velocity values predicted by our MD-DRM almostcoincides with the values reported in the mentioned references, these results togetherwith the effect of mesh refinement are shown in Table 1. The variables that appearin this table are:

ux max, ymax: maximum horizontal velocity at the vertical middle plane of the cavityand the position where the maximum occurs.

vx max, xmax: maximum vertical velocity at the horizontal middle plane of the cavityand the position where the maximum occurs.

From Table 1 note that the accuracy or the MD-DRM solution is remarkably goodconsidering that the mesh used here is much coarser than the 120 × 120 nonuniformgrid used by Vahl Davis in his finite difference solution.

The convergence criteria used in this work was defined as

‖unew‖ − ‖uold‖‖unew‖ ≤ 10−6

for the momentum equations and

‖Tnew‖ − ‖Told‖‖Tnew‖ ≤ 10−6

for the temperature field.

Table 1: Comparison of the results for different mesh densities and different Ranumbers.

10 × 10 20 × 20 20 × 20Ra uniform uniform non-uniform Vahl Davis [30]

103 ux max 3.580 3.600 3.610 3.649ymax 0.790 0.810 0.810 0.813

uy max 3.590 3.650 3.665 3.691xmax 0.175 0.180 0.180 0.177

104 ux max 14.400 16.100 16.100 16.219ymax 0.800 0.820 0.820 0.823

uy max 17.100 19.200 19.120 19.549xmax 0.100 0.110 0.110 0.123




3 Non-isothermal non-Newtonian Stokes flowwith viscous dissipation

For liquids with high viscosity and low thermal conductivity such as most polymers,viscous dissipation cannot be neglected because it has a strong influence on thetemperature field which, in turn, affects the thermophysical properties of the fluid.This phenomenon is particularly relevant in the polymer industry (in extrusion andinjection processes the fluid is forced to flow through channels where high powergeneration near the walls is observed [31]). Also in radiators design, where theworking fluids are usually highly viscous oils, dissipation in laminar flow regimeis important.

The behaviour of non-Newtonian fluids is strongly dependent on the viscosityvariations within the domain that are caused by the shear rate and temperature.Most non-Newtonian fluids such as polymers exhibit a viscosity that is a decreas-ing function of the shear rate. This characteristic is known as shear thinning [32].The viscosity of an inelastic non-Newtonian fluid can be calculated on the onehand in terms of the shear rate through several mathematical models such as thepower law model, the Carreau model and the hyperbolic tangent model [32, 33].On the other hand, the viscosity of polymer melts changes drastically with tem-perature (a change of 1% in temperature can cause at least a 25% change inviscosity) and it can be calculated by either the Andrade law [32], or the WLFequation [32]. Since the viscosity of non-Newtonian fluids is dependent on shearrate and temperature, it follows that the mathematical models governing the flowmotion are nonlinear even in the case when inertia effects can be neglected, i.e. atlow Reynolds number. This type of flow process requires sophisticated numericalsolution techniques.

Only a few works have been reported in the literature on the BEM numeri-cal solution of inelastic non-Newtonian flows. Phan-Thien [34] proposed a BEMsolution of the non-homogeneous momentum equations based on the particularsolution approach and RBF interpolation for the nonlinear terms. Davies [31] usedthe pseudo-body force technique and cell integration to model polymers and opti-mize mixing equipment. Davies also pointed out that the application of the DRMto the whole domain does not produce accurate results especially when there arehigh viscosity gradients.

As mentioned before, the cell integration BEM combined with subdomain tech-niques such as the Green’s element method GEM [6] have been applied togetherwith a velocity–vorticity formulation by Skerget and Samec [35], to model non-Newtonian flows in enclosures. As in other previous works by Skerget [36] forthe Navier–Stokes equations, the integral formulation keeps the domain integralswithout any further simplification.

In this section, we will introduce the MD-DRM solution of the coupled systemof Stokes equations and the energy equation for non-Newtonian fluids with viscousdissipation effects. Besides the efficiency and accuracy of the proposed numericalmethod, the MD-DRM has the advantage of the robust RBF interpolation used




in the DRM. Due to their smoothness or noise minimization character, the radialfunctions guarantee high accuracy on the evaluation of the gradient of the fieldvariables [10]. This property of the DRM interpolation permits a precise evaluationof the velocity and temperature derivatives directly from the solution of the matrixsystem without any additional burden.

3.1 Governing equations

The system of mass, momentum and energy conservation equations for the steady-state, non-isothermal flow of an incompressible fluid including buoyancy andviscous dissipation is given in tensor notation by

∂ui

∂xi= 0, x ∈ � (59)

− ∂p

∂xi+ ∂

∂xj(ηεij) + ρgiβ(T − Tc) = 0, x ∈ � (60)

k∂2T

∂xj∂xj+ σijεij = ρCpuj

∂T

∂xj, x ∈ � (61)

with boundary conditions

ui = ui0, x ∈ �u (62)

ti = σijnj = ti0, x ∈ �t (63)

T = T0, x ∈ �T (64)

q = −k∂T

∂n= q0, x ∈ �q (65)

where εij = 12 (∂ui/∂xj + ∂uj/∂xi) is the strain rate tensor, σij the total stress tensor,

ui is the velocity vector, ti is the traction vector, ni is the outward unit normal tothe boundary � = �u + �t = �T + �q of the volume�, T is the fluid temperatureat any point, p the pressure and q is the heat flux. The thermophysical propertiesare: ρ the fluid density, η the viscosity which is a function of temperature andthe generalized strain rate, k is the thermal conductivity, Cp the thermal capacityand β is the coefficient of thermal expansion. The buoyancy force ρgiβ(T − Tc)in eqn (60) represents the effect of temperature change on density, where Tc is thereference temperature and gi is the magnitude of gravity acting in the i direction.The term σijεij in eqn (61) is the irreversible rate of internal energy increase perunit volume by viscous dissipation or in other words the degradation of mechanicalto thermal energy.

To convert this problem into a perturbation to a base Newtonian flow, the totalstress tensor is decomposed into a Newtonian and a nonlinear component as follows,

σij = −pδij + ηNεij + τ (e)ij (66)




where ηN is an arbitrary constant viscosity that can be chosen to be the zero shearrate viscosity and τ (e)

ij represents the non-Newtonian effects in the stress tensor. Forinelastic generalized Newtonian fluids the nonlinear terms are:

τ(e)ij (γ ) = (η − ηN )εij (67)

where the non-Newtonian viscosity η is a function of the generalized shear rate γgiven by

γ = √2εijεji (68)

With the formulation introduced in eqn (66) the momentum equation (60) canbe rewritten as

− ∂p

∂xi+ ηN

∂2ui

∂xj∂xj+ ρgiβ(T − Tc) + ∂τ

(e)ij

∂xj= 0 (69)

The viscosity of most non-Newtonian fluids such as polymers is usually a decreas-ing function of the generalized shear rate γ and this is known as shear-thinningbehaviour. For a Newtonian fluid the viscosity η is a constant value µ,

η(γ ) = µ = constant (70)

On the other hand, the most commonly used expression for the viscosity of a non-Newtonian fluid is the power law or Ostwald–de-Waele model [32],

η(γ ) = K γ n−1 (71)

where K is called the consistency index and n ∈ [0, 1], the power law index.There are other more accurate semi-theoretical models to describe the behaviour

of non-Newtonian fluids, some of them are:

• The Carreau model [32]

η(γ ) = η∞ + η0 − η∞[1 + (λγ )2](1−n)/2 (72)

This model has four adjustable parameters to fit the experimental data: λ is acharacteristic time, η∞ is a constant viscosity at very high shear rates, η0 and nare the same as in the power law model.

• The hyperbolic tangent model [32]

η(γ ) = A − B tanh

(γ

k

)n

(73)

where the parameters A, B, k and n are obtained by data fitting techniques toexperimental data.




These models account for the effect of the shear rate on viscosity. However, theactual viscosity is also dependent on temperature through the Andrade law [32,37]of the form

η = η(γ )eER

(1T − 1

T0

)(74)

where E is the activation energy of the fluid; R the ideal gas constant and η(γ ) is theviscosity evaluated at the reference temperature T0. Besides, in the above system ofequations the flow field and temperature field are completely coupled, i.e. two-waycoupling, making the problem even more complex.

3.2 Multi-domain integral formulation

In terms of the Stokes fundamental solution, the velocity field in this problem hasthe same integral representation formulae as the previous one, i.e. eqn (16), butwith the vector density g in the domain integral given by

gi = −ρgiβ(T − Tc) − ∂τ(e)ij

∂xj(75)

instead of the convective and bouyancy terms in (17)Similarly, the integral representation of the energy transport equation in terms of

the Laplace’s fundamental solution, at each domain, is given by eqn (23) with thescalar density h in the domain integral given by

h = uj∂T

∂xj− σijεij (76)

instead of the convective term given in (25).As before, the final set of equations is completed by assembling the integral

equations for each domain element, using the traction equilibrium and velocitycompatibility at the common interfaces between subregions, i.e. u(+)

i = u(−)i and

σ(+)ij nj + σ (−)

ij nj = 0. Likewise, the temperature integral equations must also beassembled using the continuity of the temperature field and the heat flux balancecondition, i.e. T (+)

i = T (−)i and ((∂T/∂xj)(+))nj + ((∂T/∂xj)(−))nj = 0.

To express the domain integral in equations in terms of equivalent boundaryintegrals, the DRM approximation is reemployed, as explained before. The evalu-ation of the extra stress tensor τ (e)

ij and other terms appearing in eqns (75) and (76),require the numerical approximation of derivatives of velocity and temperature,which are evaluated using the approach given in Section 2.4. Once these deriva-tives are known, they can be used to obtain the value of the non-Newtonian viscosity,the stress tensor and the convective terms at each point of each subdomain.

For the numerical solution of the problem the surface �n of each sub-region or domain element can be discretized by means of isoparametric linear




boundary elements. Along each element the integrals are calculated in termsof the nodal values of the velocity and tractions and using linear interpolationfunctions.

In the present case, the resulting system of integral equations are nonlinear andcoupled through several temperature terms. Their corresponding matrix represen-tation can be solved using a Newton–Raphson scheme combined with a line-searchalgorithm intended to reduce the error at each iteration [25].

To account for the coupling between the momentum and energy equations, weused direct iteration. The velocity in the convective term from the energy equationis considered as a known variable from the previous solution of the momentumequations obtained by the Newton–Raphson method. Likewise, at each iterationthe temperature-dependent terms in the momentum equations can be known fromthe previous solution of the energy equation.

For fluids with variable viscosity, there is an additional source of problems inthe numerical iteration. In the multi-domain solution of non-Newtonian problemsthere will be subdomains where the viscosity remains nearly constant and otherregions with higher gradients where the viscosity changes ostensibly. These differ-ences between subregions can make the residual function of the iterative methodhave flat regions of local minima or valleys where the iteration process stagnates.Additionally, the DRM for the non-Newtonian case is based on a perturbation for-mulation given in eqn (69), so when the assumed constant viscosity ηN is very farfrom its actual value η, Newton’s method might not make good progress unlessthe initial guess for the iterations is very close to the true solution. To alleviate theproblems mentioned above, a predictor-corrector approach was designed in whicha sequence of linear problems can be solved to get a better initial guess for startingthe nonlinear iterations. In the predictor stage each subdomain is assumed to have aconstant viscosity equal to the average viscosity of the boundary nodes that definethe subdomain. This average value is then corrected in the corrector stage of theprocess when the original nonlinear system of equations is solved by Newton’smethod until convergence is reached. If Newton’s method stops at a local mini-mum, the predictor step can be performed once again but starting from that localminimum and so on.

3.3 Non-isothermal Couette flow with viscous dissipation

In this example we consider the flow of an incompressible power law fluid betweentwo axial cylinders as shown in Fig. 2.As the inner cylinder rotates, each cell of fluidrubs against the adjacent cells. This rubbing of adjacent layers of fluid producesheat; that is, mechanical energy is degraded into thermal energy. The magnitude ofthe viscous dissipation effect depends on the local velocity gradient. The surfacesof the inner and outer cylinders are maintained at the same temperature T = T0.The geometry is defined by the radius R1 = 1 m and R2 = 5 m.

In our numerical simulation of this problem, for the MD-DRM subdomainapproach, a subdivision of 400 domain elements mesh was used as shown in Fig. 3.




1R

2R

ω

0T

0T

Figure 2: Couette flow with viscous heat generation.

The mathematical model for this example consists of the continuity equation, themomentum equations and the energy equation, i.e.

∂ui

∂xi= 0, x ∈ � (77)

− ∂p

∂xi+ ∂

∂xj(ηεij) = 0, x ∈ � (78)

k∂2T

∂xj∂xj+ σijεij = 0, x ∈ � (79)

There are three possible cases depending on whether the viscosity is dependent ontemperature, on shear rate or on both of them. In all these cases the problem consistsof a two-way coupling system of nonlinear equations.

3.3.1 Case I: Viscosity does not vary with temperature η = η(γ )For a power law fluid with a viscosity that is only a function of the generalizedshear rate the rheological model is given by eqn (71) in terms of the two constantsK = 1 Pa · sn (the consistency coefficient) and n (power law index). The numericalresults for different values of the parameter n can be seen in Figs 4–6.




Figure 3: Uniform 20 × 20 domain elements mesh used for MD-DRM solution ofthe Couette problem.

Figure 4 presents our result for the different velocity profiles, Fig. 5 the viscosityprofiles and Fig. 6 the temperature profiles. These results have been comparedwith the analytical solution obtained according to [33] by direct integration of themomentum and energy equations. In all cases the numerical error was less than0.5%, as shown in the figures.

3.3.2 Case II: Viscosity is a function of temperature only η = η(T )In this case we have considered that viscosity is constant (Newtonian fluid) and thetemperature rise in the fluid results in variations of the viscosity which then affectthe velocity profile. The equations of motion and energy are coupled and must besolved by an iterative method for each point of the flow system. The expression forthe viscosity in this case is the following (see eqn (74))

η = η0eER

(1T − 1

T0

)(80)

with η0 = 1 Pa · s.




0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

r

u

n=1

n=0.8

n=0.6

n=0.4

− Analyticalo MD−DRM

Figure 4: Velocity profiles for the Couette flow problem with viscous dissipation.

0.5 1 1.5 2 2.50

2

4

6

8

10

12

r

µ (V

isco

sity

)

n=0.4

n=0.6

n=0.8

n=1

− Analyticalo MD−DRM

Figure 5: Viscosity profiles for the Couette flow problem with viscous dissipation.




0.5 1 1.5 2 2.548

50

52

54

56

58

60

r

T

n=1.0

n=0.8

n=0.6

n=0.4

o MD−DRM − Theoretical

Figure 6: Temperature profiles for the Couette flow problem with viscous dissipa-tion. In this problem the viscosity is constant and it does not change withtemperature.

In Figs 7–9, we show the comparison between our numerical solution for dif-ferent values of the activity energy coefficient � and the solution obtained witha completely different numerical scheme for the velocity, temperature and vis-cosity profiles, respectively. As before, the agreement between the two solutionsis excellent. The numerical scheme used to validate our result was based on afinite difference three-stage Lobatto IIIa formula which is basically a collocationtechnique [38, 39].

3.3.3 Case III: Viscosity is a function of both shear rate and temperatureη = η(γ , T )

This is the most complex case where the effects of shear rate and temperature affectthe value of viscosity, i.e.

η = K γ n−1eER

(1T − 1

T0

)(81)

As before, in this case we compare our results with those obtained with thefinite difference method mentioned above [38, 39]. Figures 10–12 show the com-parison between the velocity, temperature and viscosity profiles obtained with thetwo approaches. The numerical results were obtained for two different values ofthe power law index n of a non-Newtonian fluid at a constant activation energyE = 107 J/mol, and as can be seen in these figures the MD-DRM results are veryclose to the test solution with a numerical difference less than 1%.




0.5 1 1.5 2 2.50

1

2

3

4

5

6

r

u

E=107

E=108

E=109

− Shampineo MD−DRM

Figure 7: Velocity profiles for the Couette flow problem with temperature dependentviscosity.

0.5 1 1.5 2 2.550

51

52

53

54

55

56

57

58

59

r

T

E=107

E=108

E=109

o MD−DRM − Shampine

Figure 8: Temperature profiles for the Couette flow problem with temperaturedependent viscosity.




0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

µE=107

E=108

E=109

o MD−DRM − Shampine

Figure 9: Viscosity profiles for the Couette flow problem with temperature depen-dent viscosity.

0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

r

u

n=0.8E=107

n=0.4E=107

o MD−DRM − VBP4C

Figure 10: Velocity profiles for the Couette flow problem with viscous dissipationand temperature dependent viscosity.




0.5 1 1.5 2 2.550

51

52

53

54

55

56

r

Tn=0.8 E=107

n=0.4 E=107

o MD−DRM− BVP4C

Figure 11: Temperature profiles for the Couette flow problem with viscous dissi-pation and temperature dependent viscosity.

0.5 1 1.5 2 2.50

2

4

6

8

10

12

14

r

µ

n=0.4 E=107

n=0.8 E=107

o MD−DRM− BVP4C

Figure 12: Viscosity profiles for the Couette flow problem with viscous dissipationand temperature dependent viscosity.




4 Conclusion

In this work a MD-DRM has been described and applied for the solution of non-isothermal Newtonian and non-Newtonian flow problems with viscous dissipation.Several other numerical examples are provided in our previous articles [26, 40],including cases of natural convection with viscous dissipation. The multi-domaintechnique is basically a domain partition method which divides the entire domaininto smaller regions. In each subregion or domain cell the integral representationformulae for the flow and temperature field are applied, and between adjacentregions the corresponding matching or continuity conditions are imposed. Thedomain integrals in each domain element are treated by the DRM approximationin terms of the most efficient interpolation functions available in the mathematicalliterature. Despite the relatively rough meshes used and most simple boundaryelements the results show convergence and high accuracy.

Despite the fact that in the MD-DRM there are internal elements and a mesh,it must not be regarded as a domain method because all the domain integralsare converted into equivalent boundary integrals in each cell or element. There-fore the proposed multi-domain method preserves the boundary-only character ofthe BEM.

The different test examples presented show the versatility and efficiency of theproposed numerical scheme for the solution of nonlinear flow problems. In thepresent article we have also extended the capabilities of the MD-DRM to non-isothermal Newtonian and non-Newtonian flow problems.

References

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CHAPTER 3 Multi-domain DRM boundary element method for the … · 2014-05-10 · CHAPTER 3 Multi-domain DRM boundary element method for the numerical simulation of non-isothermal