CHAPTER 2 Viscous Compressible Fluid Dynamics

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CHAPTER 2

Viscous compressible ﬂuid dynamics

L. Škerget & M. Hriberšek Faculty of Mechanical Engineering, University of Maribor, Smetanova, Maribor, Slovenia.

Abstract

The development of boundary element methods for computation of fluid flow is predominantly focused on the flow of incompressible fluids. This assumption pro- vides a good approximation for a wide variety of engineering applications, but to achieve a better representation of physical phenomena, a compressible fluid flow must be taken into account. The goal of the chapter is, first, to give an in depth expla- nation for the transformation of Navier–Stokes equations for compressible fluids into its velocity–vorticity formulation equivalent. In addition, the pressure equation for the velocity–vorticity formulation is derived. This is followed by integral trans- formations of the governing equations based on the use of a parabolic diffusion fundamental solution. Section 8 explains discrete models, including the subdomain type discretisation. As a test example, natural convection in a differentially heated tall enclosure is presented.

1 Introduction

Most of the studies dealing with transport phenomena are based on presuming that the fluid is incompressible and viscous, where the mass density is a constant quantity, and the velocity does not depend on the mass density. Pressure in the incompressible fluid flow model is not a thermodynamic state variable, but simply a force in the linear momentum balance equation. Such an easy rheological model for the fluid is suitable for modelling of slow flows, or flows with small pressure and temperature gradients or no chemical reaction and where, therefore, the mass density differences may be neglected.

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/978-1-84564-100-9/02 34 Domain Decomposition Techniques for Boundary Elements

In this chapter we will deal with the real compressible viscous fluid with restriction to the subsonic flows where the difference in mass density significantly influ- ences the vector velocity field. However, there are no shock waves and no sudden sharp changes in the values of the field functions. In the model of compressible fluid the pressure is a thermodynamic quantity p = p(ρ , T).

2 Conservation equations

The analytical description of the motion of a continuous viscous compressible media is based on the conservation of mass, momentum, and energy with asso- ciated rheological models and equations of state [1]. The present development is focused on the laminar flow of compressible isotropic fluid in solution region bounded by boundary . The field functions of interest are the velocity vector field vi(rj, t), the scalar pressure field p(rj, t), the temperature field T(rj, t) and the field of mass density ρ(rj, t). The mass, momentum, and energy conservation equations are given by

∂ρ ∂ρvj + = 0, (1) ∂t ∂xj

∂ρvi ∂ρvivj ∂σij + = + ρgi,(2) ∂t ∂xj ∂xj ∂ρT ∂ρTvj ∂qj cp + =− + S + , (3) ∂t ∂xj ∂xj in the Cartesian frame xi, where ρ and cp denote variable fluid mass density and isobaric specific heat capacity per unit mass, t is time, gi is gravitational acceleration vector, σij represents the components of the total stress tensor, qi is specific heat diffusion flux, while S stands for the heat source term and is a Rayleigh viscous dissipation function which stands for the conversion of mechanical energy to heat and acts as an additional heat source. With the definition of Stokes material derivative of the variable (·), as given by D (·) /Dt = ∂ (·) /∂t + vk∂ (·) /∂xk, the Navier–Stokes equations are given by

∂vj 1 Dρ + = 0, (4) ∂xj ρ Dt

Dvi ∂σij ρ = + ρgi, (5) Dt ∂xj

DT ∂qj c =− + S + , (6) Dt ∂xj where c denotes speciﬁc heat per unit volume, c = cp ρ. If the incompressible ﬂuid model is applied, than the total rate of mass density variation is identically zero Dρ =0, (7) Dt

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Viscous Compressible Fluid Dynamics 35 and the mass conservation is satisﬁed with a simple restriction condition for the solenoidal velocity vector ∂vj =0. (8) ∂xj The set of eqns (4)–(6) represents an unclosed system of partial differential equations that has to be closed and solved in conjuction with appropriate rheological equations, equations of state and boundary, as well as initial conditions of the problem. The Cauchy total stress tensor σij can be decomposed into a pressure contribution p plus an extra deviatoric stress tensor ﬁeld function τij

σij =−pδij + τij, (9) where δij is the Kronecker delta. The Rayleigh viscous dissipation term is given by

∂vi = τij . (10) ∂xj

3 Linear gradient type of constitutive models

In the general Reiner–Rivlins rheological model of viscous shear ﬂuid the deviatoric stress tensor reads as τij = αδij + βε˙ij + γ ε˙ikε˙kj. (11)

By considering the equalities α =−2ηε˙ii/3 and β = 2η, the following form of the constitutive model for compressible viscous shear ﬂuid may be written 2 τij = 2ηε˙ij − η Dδij, (12) 3 where D = div v =˙εii represents the divergence of the velocity ﬁeld. Considering the eqn (10) we may write ∂v ∂v ∂vj ∂v 2 = η i i + i − ηD2. (13) ∂xj ∂xj ∂xi ∂xj 3 In the case of intensive unsteady heat transfer, it is important to take into account a terminal velocity of a moving temperature front, namely

∂T ∂qi qi =−k − λ , (14) ∂xi ∂t where material constants k and λ are the heat conductivity and the heat relaxation time. For most heat transfer problems of practical importance, the simpliﬁcation known as the Fourier law of heat diffusion is accurate enough, namely ∂T qi =−k . (15) ∂xi

4 Primitive variables formulation

Combining constitutive models for stress tensor or momentum ﬂux tensor and heat diffusion ﬂux, eqns (12) and (15) in conservation eqns (5) and (6), the following system of nonlinear equations is developed

∂vj 1 Dρ = D =− , (16) ∂xj ρ Dt Dvi ∂ 2 ∂p ρ = 2η ε˙ij − η Dδij − + ρgi, (17) Dt ∂xj 3 ∂xi DT ∂ ∂T c = k + S + . (18) Dt ∂xj ∂xj Because of analytical reasons in developing the velocity–vorticity formulation of governing equations, the operator div τ is worth writing in an extended form ∂ 2 div τ = η Lij + Lji − Dδij ∂xj 3

∂η ∂Lij ∂Lji 2 ∂ηD = 2 ε˙ij + η + η − , (19) ∂xj ∂xj ∂xj 3 ∂xi where the term ∂Lji/∂xj is equal to the grad D, due to the continuity equation, div v = D. Thus the following relation is valid ∂Lji ∂ ∂vj ∂D = = . (20) ∂xj ∂xi ∂xj ∂xi ˙ By considering the equality between tensor ﬁeld functions ε˙ij = Lij − ij, one can derive an expression

∂η ∂η ˙ ∂Lij ∂D 2 ∂ηD div τ = 2 Lij − 2 ij + η + η − . (21) ∂xj ∂xj ∂xj ∂xi 3 ∂xi Finally, substituting the equalities

2 ∂Lij ∂ vi ∂η ˙ ∂η = and 2 ij =−eijk ωk, (22) ∂xj ∂xj∂xj ∂xj ∂xj the ﬁrst extended form of div τ may be formulated as follows 2 div τ = η v + grad η × ω + 2 grad v · grad η + η grad D − grad (ηD). (23) 3 By the derivation of the second extended form we also take into account the equalities div (grad v ) = grad D − rot ω = v, (24) rot (ηω ) = η rot ω − ω × grad η, (25)

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Viscous Compressible Fluid Dynamics 37 obtaining the second extended form

4 div τ =−rot(ηω ) + 2grad η × ω + 2grad v · grad η + grad (ηD) 3 − 2D grad η. (26)

Considering the second extended form of the term div τ , eqn (26), the momentum equation (17) may be written in a form suitable to derive the velocity–vorticity formulation, e.g. in a vector form

Dv 4 ρ =−rot(ηω ) + 2grad η × ω + 2grad v · grad η + grad (ηD) − 2D grad η Dt 3 − grad p + ρg , (27) or in Cartesian tensor formulation

Dvi ∂ηωk ∂η ∂η ∂vi 4 ∂ηD ∂η ρ =−eijk + 2eijk ωk + 2 + − 2D Dt ∂xj ∂xj ∂xj ∂xj 3 ∂xi ∂xi ∂p − + ρgi. (28) ∂xi Representing the material properties of the ﬂuid, e.g. the dynamic viscosity η, heat conductivity k, the speciﬁc heat per unit volume c, and the mass density ρ,as a sum of a constant and a variable part ! η = ηo + !η, k = ko + k, c = co +!c, ρ = ρo + ρ!, (29) then the momentum and energy equations (27) and (18) can be written in analogy to the basic conservation equations formulated for the constant material properties

Dv 1 ρ 1 m =−νo∇× ω − ∇ p + g + f , (30) Dt ρo ρo ρo DT S Sm = ao T + + + , (31) Dt co co co with the pseudo-body force term f m and the pseudo-heat source term Sm introduced into the momentum equation (30) and into the energy equation (31), respectively, capturing the variable material property effects, and given by the expressions

4 f m =−∇× (!ηω ) + 2∇ η × ω + 2 ∇ v · ∇ η + ∇ (ηD) − 2D∇ η − ρ!a , (32) 3 or in tensor notation form as ! ! m =− ∂ηωk + ∂η + ∂η ∂vi + 4 ∂ηD − ∂η − ! fi eijk 2eijk ωk 2 2D ρai, (33) ∂xj ∂xj ∂xj ∂xj 3 ∂xi ∂xi

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 38 Domain Decomposition Techniques for Boundary Elements which for plane ﬂow problems reduces to the relation ! ! m =− ∂ηω + ∂η + ∂η ∂vi + 4 ∂ηD − ∂η − ! fi eij 2eij ω 2 2D ρai, (34) ∂xj ∂xj ∂xj ∂xj 3 ∂xi ∂xi while the pseudo-heat source term is given by an expression

DT Sm = ∇ (!k ∇ T) −!c , (35) Dt in which the kinematic viscosity is νo = ηo/ρo, the heat diffusivity ao = ko/co, and the inertia acceleration vector is a = Dv/ Dt.

5 Velocity–vorticity formulation

The divergence and the curl of a vector ﬁeld function are the basic differential operators in vector analysis. Written for the velocity ﬁeld vi(rj, t) they represent a local expansion rate D

∂vj 1 Dρ div v = = D, D =− (36) ∂xj ρ Dt and the local vorticity vector ωi(rj, t)

∂vk ∂ωj rot ω = eijk = ωi, = 0, (37) ∂xj ∂xj representing a solenoidal vector by definition, the fluid motion computation procedure may be partitioned into its kinetics and kinematics.Vorticitydefinition equation (37) and the continuity equation (36) represent a differential description of the kinematic aspect of viscous compressible fluid motion. The kinematics deals with the relationship among the velocity field at any given instant of time, and the vorticity and mass density fields at the same instant. If the velocity and mass density fields are known in the solution domain, the corresponding vorticity field can be estab- lished through eqn (37). For the known vorticity and mass density field functions, the corresponding velocity vector can be determined by solving eqns (36) and (37), provided that appropriate boundary conditions for the velocity are prescribed. The kinetic aspect of the fluid motion is governed by the vorticity transport equation describing the redistribution of the vorticity in the fluid domain through the various transport phenomena.

5.1 Velocity vector equation

By applying the curl operator to the vorticity deﬁnition equation (37)

∇× ω = ∇× (∇× v) = ∇ (∇· v) − v , (38)

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Viscous Compressible Fluid Dynamics 39 and by using the continuity equation (36), the following vector elliptic Poisson equation for the velocity vector is obtained

v + ∇× ω − ∇ D = 0, (39) or in tensor notation form

2 ∂ vi ∂ωk ∂D + eijk − = 0. (40) ∂xj∂xj ∂xj ∂xi

The eqn (40) represents the kinematics of a compressible fluid motion, expressing the compatibility and restriction conditions among velocity, vorticity, and mass density field functions. To accelerate the convergence and the stability of the coupled velocity, vorticity, and pressure computational iterative scheme, the false transient approach may be used for the elliptic velocity Poisson equation. By adding the artificial accumulation term, the eqn (40) can be written in its parabolic diffusion form

2 ∂ vi 1 ∂vi ∂ωk ∂D − + eijk − = 0, (41) ∂xj∂xj α ∂t ∂xj ∂xi where α is the relaxation parameter controlling the diffusion and accumulation processes. It is obvious that the governing velocity equation is exactly satisﬁed only at the steady state of the artiﬁcial transient (t →∞), when the false time derivative vanishes. For the two-dimensional plane motion the eqn (41) reduces to the relation

2 ∂ vi 1 ∂vi ∂ω ∂D − + eij − = 0. (42) ∂xj∂xj α ∂t ∂xj ∂xi

5.2 Vorticity transport equation

Substituting the convection term by the following equality

1 (v · ∇ )v = ∇ v2 − v × ω, (43) 2 the momentum equation (30) can be rewritten as

∂v ρ 1 m + ω × v =−νo∇× ω − ∇h + g + f , (44) ∂t ρo ρo

2 where h = p/ρo + v /2 is the total pressure. Finally, the vorticity transport equation is obtained by applying the curl differential operator to both the sides of eqn (44)

∂ω ρ 1 m = ∇×(v × ω) − νo∇×∇× ω + ∇× g + ∇×f , (45) ∂t ρo ρo

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 40 Domain Decomposition Techniques for Boundary Elements bearing in mind the following vector relations

∇× ∇× ω = ∇ (∇· ω) −∇2ω , (46) and

∇× (v × ω) = v(∇· ω) − (v · ∇ )ω − ω(∇· v) + (ω · ∇ )v , (47) where div ω = 0 and div v = D due to the vorticity deﬁnition and continuity equation, rendering

Dω 2 1 1 m = νo∇ ω + (ω · ∇)v − ωD + ∇×ρg + ∇×f , (48) Dt ρo ρo or in Cartesian tensor notation form

2 m Dωi ∂ ωi ∂vi 1 ∂ρgk 1 ∂fk = νo + ωj − ωiD + eijk + eijk . (49) Dt ∂xj∂xj ∂xj ρo ∂xj ρo ∂xj

On the other hand, considering the vector properties div ω = 0 and div v = D, another form of eqn (49) may be written, namely

2 m ∂ωi ∂vjωi ∂ ωi ∂ωjvi 1 ∂ρgk 1 ∂fk + = νo + + eijk + eijk . (50) ∂t ∂xj ∂xj∂xj ∂xj ρo ∂xj ρo ∂xj

For the two-dimensional plane ﬂow, the vorticity vector ω (rj, t) has just one component perpendicular to the plane of the ﬂow, e.g. ω = (0, 0, ω), and it can be treated as a scalar quantity ω. The vortex-twisting transport term is identically zero due to (ω · ∇ )v = 0, reducing the vector vorticity equation to a scalar one for vorticity ω

2 m ∂ω ∂vjω ∂ ω 1 ∂ρgi 1 ∂fi + = νo − eij − eij . (51) ∂t ∂xj ∂xj∂xj ρo ∂xj ρo ∂xj The vorticity transport statement equation (50) is a highly nonlinear partial differential equation due to the products of velocity v (rj, t) and vorticity ω (rj, t)in convective and in stretching-twisting terms, and the velocity field function is kinematically dependent on vorticity. Because of this inherent nonlinearity, the kinetics of general viscous motion, and what is drastically true for high Reynolds number flows, represents greater numerical efforts than that considered by the kinematics. Due to the buoyancy force and variable material property terms acting as additional temperature and pressure dependent vorticity source terms, the vorticity transport equation is coupled to the energy and pressure equations, making the numerical solution procedure very severe. The dilatation and the vortical part of the flow, D and ω field functions, have to be under relaxed to achieve the convergence of the numerical solution procedure.

6 Pressure equation

In compressible ﬂuid dynamics the pressure is a temperature and mass density dependent thermodynamic quantity. Let us start with the momentum equation (30) for the pressure gradient grad p

m ∇ p = fp =−ηo∇× ω − ρoa + ρg + f . (52)

In an indicial tensor formulation this equation takes the form

∂p = =− ∂ωk − + + m fpi ηoeijk ρoai ρgi fi , (53) ∂xi ∂xj which simpliﬁes in planar ﬂow case to the following dependence

∂p = =− ∂ω − + + m fpi ηoeij ρoai ρgi fi , (54) ∂xi ∂xj

where a = Dv/ Dt represents the acceleration vector. In the vector function fp the inertia, diffusion, and gravitational effects are incorporated as well as the effects of nonlinear material properties. To derive the pressure equation, depending on known ﬁeld and material functions, the divergence of eqn (52) should be calculated, resulting in the elliptic Poisson pressure equation

p − ∇·fp = 0, (55) or in a indicial tensor notation as

∂p ∂fpi − = 0. (56) ∂xi∂xi ∂xi

The Neumann boundary conditions for the pressure equation may be determined for the whole solution domain and the following relation is valid

∂p = f · n on . (57) ∂n p

By adding the false pressure transient term, the eqn (56) may be formulated as a false parabolic diffusion equation

∂p 1 ∂p ∂fpi − − = 0, (58) ∂xi∂xi α ∂t ∂xi where α is a relaxation parameter. It is obvious that the elliptic pressure equation is exactly satisﬁed only at the false transient steady state, when the false pressure accumulation term vanishes.

7 Boundary-domain integral equations

7.1 Preliminary comments

The unique property and advantage of the boundary element method originates from the application of the Green fundamental solutions as particular weighting functions [2,3]. Since fundamental solutions only consider the linear transport phenomenon, an appropriate selection of a linear differential operator L [·] is of prime importance in establishing a stable and accurate singular integral representation corresponding to the original differential conservation equation. All differential conservation models of different ﬂow ﬁeld functions can be written in the following general form

L [u] + b = 0, (59) where the operator L [·] can be either elliptic or parabolic, u(rj, t) is an arbitrary ﬁeld function, and b(rj, t) is applied for nonlinear transport effects or pseudo-body forces.

7.2 Integral representation of ﬂow kinematics

Employing the linear parabolic diffusion differential operator

∂2 (·) ∂ (·) L [·] = α − , (60) ∂xj∂xj ∂t the following relationship may be obtained

2 ∂ vi ∂vi L [vi] + bi = α − + bi = 0. (61) ∂xj∂xj ∂t

The singular boundary-domain integral representation for the velocity vector can be formulated by using the Green theorems for scalar functions or weighting residuals technique rendering the following vector integral formulation, e.g. written in the time incremental form for the time step t = tF − tF−1 tF tF ∂v c (ξ) v (ξ, tF) + α v q dtd = α u dtd t − t − ∂n F 1 F 1 tF + + bu dtd vF−1uF−1d, (62) tF−1 where u stands for the parabolic diffusion fundamental solution of the equation

∂2u ∂u α + + δ(ξ, s)δ(tF, t) = 0, (63) ∂xj∂xj ∂t

given by the expression

1 − 2 u = e r /4ατ , (64) (4πατ)d/2

and q is its normal derivative, e.g. q = ∂u/∂n = q · n. The vector flux variable = = is defined as q grad u or qi ∂u /∂xi, where (ξ, tF) and (s, t) are used for the source and reference field points, respectively, d is the dimension of the problem in τ = tF − t. Assuming a constant variation of all field functions within the individual time increment t, the time integrals in eqn (62) may be evaluated analytically, e.g. one can substitute tF tF U = α udt, Q = α qdt, (65) tF−1 tF−1

and the integral representation, eqn (62), may be rewritten as ∂v 1 c (ξ) v (ξ, t ) + v Qd = Ud + bUd F ∂n α + vF−1uF−1d. (66)

Equating the pseudo-body force with the rotational and compressible ﬂuid ﬂow part

b = α∇× ω − α∇ D, (67)

renders an integral formulation ∂v c (ξ) v (ξ, t ) + v Qd = Ud + (∇× ω)Ud F ∂n − ∇ + ( D)U d vF−1uF−1d. (68)

The pseudo-body force domain integral involves derivatives of the vorticity and dilatation ﬁelds, which can be eliminated by using the Gauss divergence theorem. Applying the vector equalities

(∇× ω)U = ∇× (ω U) + ω × ∇ U, (69) (∇ D)U = ∇ (DU) − D∇ U, (70)

∇ = where the grad U or U can be equated to ﬂux vector Qi ∂U /∂xi, the domain integrals can be written as sums (∇× ω)Ud = ∇× (ω U)d + ω × Q d =− ω × nUd + ω × Q d, (71) (∇ D)Ud = ∇ (DU)d − DQ d = DnU d − DQ d. (72) The following form of the integral representation for the kinematics is obtained as follows ∂v c (ξ) v (ξ, t ) + v Qd = Ud − ω × nUd + ω × Q d F ∂n − + + DnU d DQ d vF−1uF−1d, (73) or in tensor symbolic notation as ∂v c (ξ) v (ξ, t ) + v Qd = i Ud − e ω n Ud + e ω Qd i F i ∂n ijk j k ijk j k − + + DniU d DQi d vi,F−1uF−1d. (74) The kinematics of plane motion is given by two scalar equations as follows ∂v c (ξ) v (ξ, t ) + v Qd = i Ud + e ωn Ud − e ωQd i F i ∂n ij j ij j − + + DniU d DQi d vi,F−1uF−1d, (75) or in an extended form for the x and y components of the velocity vector ∂v c (ξ) v (ξ, t ) + v Qd = x Ud + ωn Ud − ωQd x F x ∂n y y − + + DnxU d DQxd vx,F−1uF−1d, (76) ∂vy c (ξ) v (ξ, t ) + v Qd = Ud − ωn Ud + ωQd y F y ∂n x x − + + DnyU d DQyd vy,F−1uF−1d. (77)

The eqn (74) represents the kinematics of viscous compressible fluid motion or the compatibility and restriction conditions among vorticity, velocity, and dilatation field functions in an integral form. The most important issue in numerical modelling of compressible fluid flow circumstances is to obtain a divergence-free final solution for the vorticity vector and to satisfy the mass conservation equation [4]. Thus, the kinematic integral representation should preserve the compatibility and restriction conditions for the velocity and vorticity field functions. By using additional compatibility and restriction requirements, e.g. ω = rot v and div v = D, a more convenient integral representation may be derived [5, 6]. Let us for a moment, due to derivation simplicity and clarity, focus our atten- tion to the plane two-dimensional flow kinematics given by the eqn (75). By using the expressions for the velocity components’ normal derivatives, vorticity definition, and unit tangent and normal vector, e.g. ∂vi/∂n = ∂vi/xj nj, ω = eij∂vj/∂xi =

∂vy/∂x − ∂vx/∂y, n = (nx, ny), and t = (tx, ty) = ( − ny, nx), for i, j = 1, 2, and by applying the continuity equation (16), the following relation may be derived

∂v ∂vj i + e ωn − Dn =−e . (78) ∂n ij j i ij ∂t

Thus, the boundary integrals on the right-hand side of the eqn (75) can be rewritten, resulting in the following integral formulation ∂vj c (ξ) v (ξ, t ) + v Qd =−e Ud − e ωQd i F i ij ∂t ij j + + DQi d vi,F−1uF−1d. (79)

The eqn (79) can be again reformulated as ∂vjU c (ξ) v (ξ, t ) + v Qd =+e v Qd − e d i F i ij j t ij ∂t − + + eij ωQj d DQi d vi,F−1uF−1d, (80)

= where we called the tangential derivative of the fundamental solution Qt ∂U /∂t, and by applying the Gauss theorem, the second boundary integral on the right-hand side of equation vanishes, e.g. as follows ∂vjU ∂vjU ∂vjU d = − n + n d ∂t ∂x y ∂y x ∂ ∂vjU ∂ ∂vjU = − + d ≡ 0, (81) ∂y ∂x ∂x ∂y

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 46 Domain Decomposition Techniques for Boundary Elements resulting in the ﬁnal integral representation for the two-dimensional plane kinematics + = − c (ξ) vi (ξ, tF) viQ d eij vjQt d eij ωQj d + + DQi d vi,F−1uF−1d. (82) The boundary integral representation for the general ﬂow situation can now be easily stated as, e.g. in the compact symbolic notation form for the cyclic combination of the indices, ijkij = 12312, + = − − − c (ξ) vi (ξ, tF) viQ d vk Qkni Qi nk d vj(Qi nj Qj ni)d + − + + ωjQkd ωkQj d DQi d vi,F−1uF−1d, (83) or in the form of parabolic integral vector formulation c (ξ) v (ξ, tF) + v Q d = Q × n × vd + ω × Q d + + DQ d vF−1uF−1d. (84) The vector integral statement, eqn (84), represents three scalar equations for individual x, y, z Cartesian coordinate directions, but only two of them are independent + = − c (ξ) vx (ξ, tF) vxQ d vz(Qz nx Qxnz)d − − + − vy(Qxny Qynx)d ωyQz d ωzQyd + + DQxd vx,F−1uF−1d, + = − c (ξ) vy (ξ, tF) vyQ d vx(Qxny Qynx)d − − + − vz(Qynz Qz ny)d ωzQxd ωxQz d (85) + + DQyd vy,F−1uF−1d, + = − c (ξ) vz (ξ, tF) vzQ d vy(Qynz Qz ny)d − − + − vx Qz nx Qxnz d ωxQyd ωyQxd + + DQz d vz,F−1uF−1d.

The boundary-domain integral representation, eqn (84), is completely equivalent to eqns (36) and (37), or eqn (40), together with velocity boundary conditions, expressing the kinematics, compatibility, and restriction conditions of a general viscous compressible fluid flow in the integral form. Boundary velocity conditions are included in the boundary integrals representing the contribution to the velocity field in the solution domain. The domain integral gives the contribution of the rotational and dilatation flow to the development of the velocity field. The last domain integral is due to the false initial conditions.A unique feature of the singular integral representation is that it enables the explicit determination of velocity components in the domain. This unique ability made it possible to confine the solution field to the vortical and dilatation region of the flow. Notice that for the irrotational and incompressible fluid flow the domain integrals vanish, and the kinematics of the potential fluid flow is given by the boundary integrals only. The compatibility between velocity, vorticity, and mass density field functions cannot exist for arbitrary specified vorticity and expansion rate distribution in a solution domain, since the vorticity field is kinematically restricted. Since the vorticity field in the interior of the fluid domain is determined uniquely from the solution of the vorticity transport equation (49), only the boundary vorticity values are subject to the kinematic restriction. In consequence, the vorticity boundary conditions are linked to the kinematic restriction given in the integral form within the domain integral. As was mentioned, the boundary vorticity values are expressed in the integral form within the domain integral, excluding a need for use of an appropriate approx- imative formula determining locally vorticity values on the boundary, which would bring some additional error in the numerical scheme employed [7]. Using this unique feature of global integral representation for boundary vorticity values, the vector equation (84) has to be written in its tangential form as follows c (ξ) n (ξ) × v (ξ, tF) + n (ξ) × v Q d = n (ξ) × Q × n × vd + n (ξ) × ω × Q d + n (ξ) × DQ d + × n (ξ) vF−1uF−1d, (86) in order to obtain an appropriate nonsingular implicit system of equations for unknown boundary vorticity or tangential velocity component values to the boundary. When the normal velocity component values to the boundary are unknown, the normal form of the mentioned equation has to be employed c (ξ) n (ξ) · v (ξ, tF) + n (ξ) · v Q d = n (ξ) · Q × n × vd + · × + · + · n (ξ) ω Q d n (ξ) DQ d n (ξ) vF−1uF−1d. (87)

The eqns (86) and (87) basically represent the application of the boundary velocity conditions given for the normal and tangent velocity component to the boundary. For the closed cavity flow problem, where the boundary represents the non- permeable and noslip solid wall on which v = 0, the eqn (84) reduces to = × + c (ξ) v (ξ, tF) ω Q d DQ d vF−1uF−1d. (88) For such flow circumstances, the boundary integral vanishes, and the velocity field in the domain is simply given by a domain integral of the vorticity and the expansion rate fields. Finally, it should be stressed, that the parabolic kinematic integral representation given by eqn (84) is an equivalent to the parabolised version of the velocity equation (42). Notice, that for a large time increment value, the kernel function U reduces to an elliptic Laplace fundamental solution, e.g. t →∞and U → u, the domain integral due to the false initial conditions vanishes, and the integral representation (84) reduces to the integral representation for the elliptic velocity equation (41) as follows c (ξ) v (ξ) + v qd = (q × n) × vd + ω × qd + Dq d, (89) where u is now the elliptic Laplace fundamental solution and q = ∂u/∂n.

7.2.1 External flow kinematics Let us consider a body sunk in an infinite domain of viscous and compressible fluid flow. Velocity in the free stream v ∞ can be introduced in the elliptic kinematic formulation equation (89), splitting the domain by two separate boundaries, dividing the solid body from the fluid, and ∞ at a very large distance, where the free stream velocity applies. The contribution of the velocity field at the infinite distance, or the influence of the irrotational fluid motion to the development of the velocity field can be determined in the following way. The boundary ∞ should be a very large spherical surface of radius R, with the solid body in the middle. Integral statement equation (89) can now be written for the whole boundary denoted by + = + ∞ occupying the infinite solution domain c (ξ) v (ξ) + v qd = (q × n) × vd + ω × qd + Dq d. + + (90)

The contribution on ∞ can be evaluated using the value ∇u = n/4πR2 " # 1 1 1 lim − (n · n) v ∞ − (n × n) × v∞ d =− v∞, (91) →∞ 2 2 4π R ∞ R R

and the boundary-domain integral equation of the external ﬂow around the immersed body, which has the surface , becomes c (ξ) v (ξ) + v q d = (q × n) × vd + ω × q d + Dq d + v∞. (92)

For the case of a nonrotating and impermeable solid surface (with v = 0), the above formulation equation (92) simplifies to c (ξ) v (ξ) = ω × q d + Dq d + v∞, (93) which may be recognised as the Biot–Savart’s law of induced velocity, which represents a relationship among the velocity, vorticity, and mass density field functions of an infinite domain. Since only the vorticity and dilatation distributions in a flow domain contribute to the velocity field, while the domain integrals in potential and incompressible flow parts vanish, e.g. (ω = 0), and (D = 0), the solution procedure or the discretisation has to be made only for the vortical and dilatation part of the whole domain.

7.3 Integral representation of ﬂow kinetics

Considering the kinetics in an integral representation, one has to take into account the parabolic diffusion convection character of the vorticity transport equation (50). With the use of the linear parabolic diffusion differential operator [8]

∂2 (·) ∂ (·) L [·] = νo − , (94) ∂xj∂xj ∂t

the vorticity equation can be formulated as a vector nonhomogeneous parabolic diffusion equation as follows

2 ∂ ωi ∂ωi L [ωi] + bi = νo − + bi = 0, (95) ∂xj∂xj ∂t

with the integral representation written in a time increment form for a time step t = tF − tF−1 tF tF ∂ωi c (ξ) ωi (ξ, tF) + νo ωiq dtd = νo u dtd t − t − ∂n F 1 F 1 tF + + biu dtd ωi,F−1uF−1d, (96) tF−1

where u stands for the parabolic diffusion fundamental solution, and α has to be replaced by νo. Assuming constant variation of all ﬁeld functions within the

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 50 Domain Decomposition Techniques for Boundary Elements individual time increment, eqn (96) is written as follows ∂ω 1 c (ξ) ω (ξ, t ) + ω Qd = i Ud + b Ud i F i ∂n ν i o + ωi,F−1uF−1d. (97)

The domain integral of the nonhomogeneous nonlinear contribution bi, represented as =−1 ∂ − − − m bi ρovjωi ρoωjvi ρeijkgk eijkfk , (98) ρo ∂xj includes the transport and source effects of the vorticity, and, therefore, we obtain the following parabolic integral representation for vorticity kinetics ∂ω c (ξ) ω (ξ, t ) + ω Qd = i Ud i F i ∂n 1 ∂ − ρ v ω − ρ ω v − ρe g − e f m Ud η ∂x o j i o j i ijk k ijk k o j + ωi,F−1uF−1d. (99) The domain integral in eqn (99) incorporates space derivatives of field functions. The problem may be sorted out in various ways, i.e. using Gaussian theorem, the derivatives are transfered to the fundamental solution, and the following integral equation is obtained ∂ω c (ξ) ω (ξ, t ) + ω Qd = i Ud i F i ∂n 1 − ρ v ω − ρ ω v − ρe g − e f m n Ud η o j i o j i ijk k ijk k j o 1 + ρ v ω − ρ ω v − ρe g − e f m Qd η o j i o j i ijk k ijk k j o + ωi,F−1uF−1d. (100) The boundary-domain integral equation (100) expresses the space vorticty transport in an integral form. Vorticity diffusion is described by the first two boundary integrals, while the third boundary integral gives the terms representing the convective vorticity flux and transport across the surface due to twisting-stretching effect, which vanish for vn = 0orv = 0 respectively, and the vorticity generation on the boundary due to buoyancy forces and variable material properties. The first domain integral gives the influence of the transport effects in the domain due to the convection, twisting-stretching, bouyancy forces, and variable material property values, while the second domain integral is due to the initial vorticity distribution effect on the development of the vorticity field in the next time interval.

7.3.1 Integral representation of plane ﬂow kinetics Governing the two-dimensional plane vorticity equation (51) is a nonhomogeneous scalar parabolic diffusion–convection partial differential equation. Applying the linear parabolic diffusion differential operator equation (94), the plane vorticity equation can be treated as a scalar nonhomogeneous parabolic diffusion equation

∂2ω ∂ω L [ω] + b = νo − + b = 0, (101) ∂xj∂xj ∂t where the nonhomogeneous term b stands for pseudo-body force or source term. Thus, the following boundary-domain integral statement corresponding to eqn (101) can be straightforwardly derived tF tF ∂ω c (ξ) ω (ξ, tF) + νo ωq dtd = νo u dtd t − t − ∂n F 1 F 1 tF + + bu dtd ωi,F−1uF−1d, (102) tF−1 where u is now a parabolic diffusion two-dimensional plane fundamental solution. Equating the body force term b to the convection, buoyancy forces, and nonliner material properties =−1 ∂ + + m b ρovjω ρeijgi eijfi , (103) ρo ∂xj the following parabolic integral formulation can be written ∂ω c (ξ) ω (ξ, t ) + ωQd = Ud F ∂n − 1 ∂ + + m + ρovjω ρeijgi eijfi U d ωF−1uF−1d, (104) ηo ∂xj where a constant variation of all field functions within the individual time increment t = tF − tF−1 is assumed. Applying the Gaussian theorem to the first domain integral of eqn (104), the direction derivatives of vorticity and density are shifted to the Green function, resulting in a boundary-domain integral formulation without derivatives of field functions within the domain integral ∂ω c (ξ) ω (ξ, t ) + ωQd = Ud F ∂n 1 − ρ v ω + ρe g + e f m n Ud η o j ij i ij i j o + 1 + + m + ρovjω ρeijgi eijfi Qj d ωF−1uF−1d. (105) ηo

The eqn (105) represents plane vorticity transport in the integral form in a physically justified manner, showing a complete analogy with eqn (100) for space fluid flow, with an exception of the twisting-stretching transport term, which appears only in the three dimensional case. To stress the physics of the transport mechanism, the eqn (105) can be rewritten as 1 ∂ω c (ξ) ω (ξ, t ) + ωQd = η − ρ v ω + ρg + f m Ud F η o ∂n o n t t o + 1 + + m + ρovjω ρeijgi eijfi Qj d ωF−1uF−1d, (106) ηo

m where vn, gt and ft are the normal velocity, and the tangential gravity and nonlinear material source components, respectively, e.g. vn = v · n, gt = g · t =−eijginj, and m = m · =− m ft f t eijfi nj. The boundary integrals describe the total vorticity flux on the boundary, due to molecular diffusion, convection, and vorticity generation by a tangential force and source terms. The first domain integral gives the influence of forced and natural convection, and nonlinear material behaviour, while the last domain integral is due to the initial vorticity distribution effect on the development of the vorticity field in a subsequent time interval.

7.4 Integral representation of pressure equation

The pressure equation (57) is a parabolic equation, and, therefore, employing the linear parabolic diffusion differential operator

∂2(·) ∂(·) L [·] = α − , (107) ∂xj∂xj ∂t the following expression can be obtained

∂2p ∂p L p + b = α − + b = 0. (108) ∂xj∂xj ∂t

The corresponding singular parabolic integral representation is given by tF tF ∂p c (ξ) v (ξ, tF) + α pq dtd = α u dtd t − t − ∂n F 1 F 1 tF + + bu dtd pF−1uF−1d, (109) tF−1 where u stands for the parabolic diffusion fundamental solution. Assuming a constant variation of all ﬁeld functions within the individual time increment t, the time integrals in eqn (109) may be evaluated analytically, and the integral representation,

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Viscous Compressible Fluid Dynamics 53 eqn (62), can be rewritten in the following form ∂p 1 c (ξ) v (ξ, t ) + pQd = Ud + bUd F ∂n α + pF−1uF−1d. (110) By equating the pseudo-body force term with the expression

∂fpi b =−α , (111) ∂xi the integral equation is given by ∂p ∂fpi c (ξ) p (ξ, t ) + pQd = Ud − Ud F ∂n ∂x i + pF−1uF−1d. (112) Using the Gaussian theorem, the domain integral in eqn (112) may be rewritten as follows ∂f pi = − U d fpiniU d fpiQi d, (113) ∂xi and, because ∂p/∂n = fp · n, the final form of the pressure integral equation is obtained + = + c (ξ) p (ξ, tF) pQ d fpiQi d pF−1uF−1d, (114) where the vector fp is given by eqn (52). For given Neumann boundary conditions, the pressure field p(rj, t) is determined to within a constant, with the solution of the parabolic differential equation (57) or the corresponding integral equation (114) taking into account the known velocity and vorticity field functions, and material property values at a given time interval. Thus, grad p is uniquely determined by the current flow field functions, independent of the flow history. The integral statement has to be first applied on the boundary for implicit evaluation of pressure boundary values. The computation of domain pressure values follows in an explicit manner.

7.5 Integral representation of heat energy kinetics

The integral representation of the nonlinear heat energy diffusion–convection transport equation is derived considering the linear parabolic diffusion differential operator ∂2(·) ∂(·) L [·] = ao − , (115) ∂xj∂xj ∂t

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 54 Domain Decomposition Techniques for Boundary Elements and, therefore, eqn (31) may be rewritten in the form

∂2T ∂T L [T] + b = ao − + b, (116) ∂xj∂xj ∂t with the following corresponding integral representation written for a time step t = tF − tF−1 tF tF ∂T c (ξ) T (ξ, tF) + ao Tq dtd = ao u dtd t − t − ∂n F 1 F 1 tF + + bu dtd Ti,F−1uF−1d, (117) tF−1 where u is the parabolic diffusion fundamental solution. Assuming constant variation of all ﬁeld functions within the individual time increment, the eqn (117) is written as follows ∂T 1 c (ξ) T (ξ, t ) + TQd = Ud + bUd F ∂n a o + TF−1uF−1d. (118)

The domain integral of the pseudo-body forces includes the effects of convection, Rayleigh dissipation function, the heat source term, and terms due to nonlinear material properties, namely 1 ∂ !∂T 1 ∂c ∂T b = k − cvjT + Tvj + cTD −!c co ∂xj ∂xj co ∂xj ∂t 1 + ( + S). (119) co

Therefore, the following integral representation can be evaluated ∂T 1 ∂ !∂T c (ξ) T (ξ, tF) + TQ d = U d + k − cvjT U d ∂n ko ∂xj ∂xj 1 ∂c ∂T 1 + Tvj + cTD −!c U d + ( + S) U d ko ∂xj ∂t ko + TF−1uF−1d. (120)

By using the Gauss divergence theorem the field function derivatives in the first domain integral are transfered to the fundamental solution, and, therefore, the following integral representation of the heat energy kinetics is obtained 1 ∂T c (ξ) T (ξ, tF) + TQ d = k − cvnT U d ko ∂n 1 ∂T 1 ∂c ∂T − !k − cv T Qd + Tv + cTD −!c Ud k ∂x j j k j ∂x ∂t o j o j + 1 + + ( S) U d TF−1uF−1d. (121) ko The boundary integrals describe the total heat flux on the boundary due to molecular diffusion and convection. The first domain integral gives the influence of the convection and the nonlinear diffusion flux, the second domain integral includes the nonlinear material effects, the third domain integral describes the Rayleigh dissipation and heat source term effects, while the last one is due to the initial temperature distribution effect on the development of the temperature field in subsequent time intervals.

8 Discrete models

After the derivation of integral equations the next step consists of transforming these equations into their algebraic forms [9–11]. The transformation is done by the means of boundary and domain discretisation and by the use of interpolation polinomials for representation of field functions. Different discrete models were developed to study the stability and convergency of the developed numerical scheme. The basic discrete model consists of a single domain discretisation, which means that the domain is treated as a single entity. The outer boundary of the domain is divided into boundary elements and the domain interior is divided into internal cells. The second discrete model is connected with the idea of subdomain technique or a multi-domain model. The subdomain technique was introduced with the aim of reducing memory and computer time demands of BEM computations. Large subdomains found their application in various areas of BEM computations, especially in linear and weakly nonlinear physical problems, i.e. heat conduction, linear elasticity, etc. In the case of computation of highly nonlinear phenomena, such as the present nonisothermal viscous compressible fluid flow, the standard subdomain technique was not efficient enough. The application of the diffusion or the diffusion–convection fundamental solution and its local implementation in parts of the computational domain resulted in the concept of a macro-element. It can be viewed as a limit version of the classical subdomain technique for BEM.

8.1 Single domain model

The single domain model has several advantages but also some severe limitations when considered in the BEM context. Among the advantages are its simple use,

as the computational mesh consists of an outer boundary element mesh (Fig. 1), whereas the interior of the domain is discretised by internal cells or by means of dual and multiple reciprocity approximations of domain integrals in the case of a weak nonlinearity only, e.g. low Peclet number value flows. The number of unknowns in the flow kinematics system matrix is defined by the number of nodes on the outer boundary of the computational domain, i.e. the system matrix is very small. Computation of field function values in the interior of the domain is done explicitly, after the boundary node values are resolved. This step is fast and can be done in full parallel mode. The disadvantages of the single domain models are the following: fully populated system matrix of both flow kinetics and flow kinematics, extremely large computing times for all the boundary as well as domain integrals, and relatively low numerical stability of the BEM algorithm when applied to highly nonlinear problems. The latter can be resolved by extending the system matrix of flow kinetics to all computational nodes, boundary as well as domain. Unfortunately, this results in a fully populated large system matrix, which is hard to store in core and solve when the computational mesh becomes more dense.

8.2 Subdomain model

The subdomain discrete model divides the computational domain into large subdomains, each consisting of a large number of boundary elements and internal cells. An example of such a mesh is presented in Fig. 2. By this approach, the large system matrix of ﬂow kinetics can be transformed into a block-type structure, allowing the use of storage reducing schemes and fast iterative solvers, [10]. The drawback of the subdomain technique in the BEM context is the need to use a mix of continuous and discontinuous boundary elements and internal cells in order to avoid problems with overdetermination of the ﬁnal system of algebraic equations,

Figure 1: Discrete model: single subdomain, boundary element mesh (left), internal cell mesh (right).

Figure 2: Discrete model: large subdomain mesh, internal cell mesh the same as in Fig. 1.

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Viscous Compressible Fluid Dynamics 57 resulting from the existence of star points. The other, more severe drawback, is the high sensibility of accuracy of flow kinematics computation, when a subdomain mesh is applied also for computation of flow kinematics. It was shown [10] that only the segmentation technique is applicable for flow kinematics in its present form. The main steps of the segmentation technique for flow kinematics can be described as: 1. Computation in original domain without interface boundaries. In this way, boundary values of unknown vorticities and velocities are computed without additional error due to additional interface boundaries. 2. After the boundary values for velocities and vorticities are computed, unknown values of velocities at points on the interface boundaries are evaluated, still in the original mesh. 3. With the additional known velocities from Step 2, computation moves to the subdomain mesh, where values of velocities inside each subdomain are computed independently of other subdomains. Another possibility is to decrease the size of a subdomain, that eventually merges with the dimension of one internal cell, resulting in a macro-element discretisation (Fig. 3). In [9], the macro-element discretisation was introduced and applied for both flow kinetics and flow kinematics, the latter in its velocity-vector form. With this approach, one macro-element or subdomain consists of one quadrilateral internal cell and four boundary elements (Fig. 4). The geometrical singularities are overcome by using discontinuous boundary elements and internal cells of different order. Constant linear discontinuous and quadratic discontinuous boundary elements were developed. Internal cells developed were constant, linear discontinuous, quadratic discontinuous, and quadratic continuous quadrilateral cells. Exten- sive testing proved that there is a need for a continuous function field at the corners of a cell, as only a combination of discontinuous quadratic boundary elements and

Figure 3: Discrete model: macro-element mesh. rrr rrr r r

rrrr

r r r r r rrr

Figure 4: Layout of a macro-element.

subdomain i subdomain j

ωij = ωji (122) ji ij ωij =− ωji n ωij • ωji -n q q (123) A or A ij ji A ω ∂ω − 1 =− ∂ω − 1 A q vnω vnω (124) AU ∂n ν ∂n ν

ij ji

Figure 5: Interface boundary conditions for vorticity transport. quadratic continuous quadrilateral cells (Fig. 4) proved to ensure the stability of the BEM numerical algorithm. When using any kind of a subdomain technique, the interface between subdomains is discretised twice, once for each subdomain. This means we have two nodal points at the same geometrical point, resulting in an increase of the overall number of unknowns in the system matrix. The nodes are linked together by application of compatibility and continuity conditions, presented in Fig. 5 for the case of ﬂow kinetics of vorticity transport. This also allows the use of heterogenous material properties in the computational domain, i.e. density, viscosity, or heat conductivity.

8.3 Hybrid approaches

The computation of flow kinematics, resulting from the velocity Poisson equation, can be combined with the macro-element computation of flow kinetics, as shown recently, [11]. Here, the segmentation technique is used to lower the computational demands of flow kinematics, and macro-element discretisation is applied in flow kinetics, resulting in sparse system and integral matrices, suitable for fast solution by the use of preconditioned iterative methods. Another approach, where the macro-element technique is used, is the mixed boundary element discretisation, presented in [12]. Its main idea is the use of macro- elements and a combination of continuous boundary elements for field functions and discontinuous boundary elements for fluxes.Although this approach does result in sparse systems of equations, its main drawback is the overdetermined system of equations.

9 Test example: differentially-heated tall enclosure

The coupled momentum energy ﬂow case in a square thermally driven cavity is frequently considered as an exercise for the incompressible ﬂow by numerical

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Viscous Compressible Fluid Dynamics 59 models with Boussinesq approximation, [9, 13, 14], in which a series of steady state reference numerical solutions for Rayleigh number values in the range 103 < Ra < 108 are presented. The study of natural convection in a non-Newtonian fluid saturated porous square cavity is presented in [15] for different Rayleigh and Darcy number values. When large temperature differences are considered, the use of a compressible numerical model solver, capable of solving the low Mach number value flows [6, 16, 17], is necessary in order to obtain physically relevant results. The natural convection inside a rectangular, differentially-heated enclosure is considered less frequently than the standard square case. The fluid motion is caused by buoyancy force due to the heated left wall while the right wall is cooled and the rest of the boundary is adiabatic. The problem has been studied in [18] specially at a near critical Rayleigh number value Ra = 3.1 × 105 inside a reactangular cavity of aspect ratio 8 : 1, when the field functions reach their quasi-steady states and oscilate around mean values, while all material properties are assumed to have constant values and the Boussinesq approximation is considered. The natural convection solutions in a differentially heated cavity of aspect ratio 4 : 1 under large temperature gradients are presented in [19]. For the constant physical properties, the transition to unsteadiness takes place with a critical Rayleigh 5 5 number value 3.2 × 10 < Rac < 3.4 × 10 . If the material properties are nonlinear functions of temperature, the transition is quite different. As reported in [19], the solutions are steady at Ra < 2 × 105, while by slightly increasing the Rayleigh number value, the steady state is obtained after a very long integration time, and for a Rayleigh number value Ra > 2.025 × 105, the solution does not seem to be steady. When the Ra > 2.3 × 105 the unsteadiness is emphasised. First the incompressible Boussinesq fluid model is assumed in the BEM numerical simulation of natural convection study in a rectangular enclosure with aspect ratio 8 : 1, Fig. 6. In this case the Rayleigh number is defined as g β(T − T ) W 3 Ra = Pr h c 2 , (125) νo where Pr is the characteristic nondimensional Prandtl number, e.g. Pr = 0.71 for air, W is a characteristic width of the enclosure, Th and Tc are the hot and the cold wall temperatures, respectively, To is a reference temperature defined as To = (Th + Tc)/2 and νo(To) is a corresponding reference kinematic viscosity. In our case we choose Th = 0.5 and Tc =−0.5 and the material properties are assumed to be constant values. The heat transfer through the wall is represented by a local and an average Nusselt number, Nu(y) and Nu¯ , respectively, defined as α(y)W W ∂T Nu(y) = = ko , (126) ko ko(Th − Tc) ∂x w 1 L Nu¯ = Nu(y)dy. (127) L 0 In the current analysis the flow patterns for Rayleigh number values Ra = 103, Ra = 104, Ra = 105, and Ra = 3.1 × 105 were studied. Only one nonuniform

∂T/∂n =0 H

Th Tc

(To,ρo)

∂T/∂n =0 x 0 W

Figure 6: Presentation of the thermally driven cavity problem in a rectangular enclosure with aspect ratio 8 : 1: boundary and initial conditions; computational mesh (10 × 60 cells with ratio 4). disctretisation was considered consisting of a M = 10 × 60 mesh of three-node quadratic boundary elements and nine-node quadratic quadrilateral internal cells with the aspect ratio of 1 : 4 in the x− and y−directions. For all cases the existence of a steady state was assumed ﬁrst and the time step value of t = 1016 was selected. The results for Ra = 103−105 are presented in Figs 7 and 8. Next, for the Ra = 3.1 × 105 ﬂow case the time dependent analysis was performed also by running the simulation from a steady-state solution for Ra = 105, used as the initial condition. The dimensionless time step value of t = 0.002 was used and the simulation results after 100 time steps are presented in Fig. 8. The representative velocity and thermal boundary layers can easily be seen on the hot and cold walls. In the second study, large temperature differences are considered in the rectangular cavity of the same dimensions as for the Boussinesq case, which impose the

Figure 7: Velocity ﬁeld and temperature isolines, incompressible ﬂow: Ra = 103, 104, left to right.

Figure 8: Velocity ﬁeld and temperature isolines, incompressible ﬂow: Ra = 105, 3.1 × 105, left to right.

WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Viscous Compressible Fluid Dynamics 63 use of the compressible numerical model solver. For a compressible ﬂuid motion the Rayleigh number is deﬁned as

g ρ2(T − T ) W 3 Ra = Pr o h c 2 , (128) Toηo where Pr is the characteristic nondimensional Prandtl number, e.g. Pr = 0.71 for air, W is the width of the cavity, Th and Tc are the hot and the cold wall temperatures, respectively, To is a reference temperature defined as To = (Th − Tc)/2, po is a reference pressure, and ρo(To, po) is a corresponding reference mass density. In our case we choose To = 600 K and po = 101,325 Pa. The temperature difference can be represented by a nondimensional parameter , defined as T − T = h c . (129) 2To The heat transfer through the wall is represented by a local and an average Nusselt number, Nu(y) and Nu¯ , respectively, defined as α(y)W W ∂T Nu(y) = = k(T) , (130) ko ko(Th − Tc) ∂x w 1 L Nu¯ = Nu(y)dy, (131) L 0 where k(T) is temperature-dependent heat conductivity and ko = k(To). In the performed test cases the Pr number value is assumed to remain constant (0.71), the temperature dependence of the viscosity is given by the Sutherland’s model η(T) T 3/2 T ∗ + S = , (132) η∗ T ∗ T + S and the heat conductivity is expressed as

η(T)cp k(T) = (133) Pr ∗ ∗ −5 with T = 273 K, S = 110.5K,η = 1.68 × 10 kg/m/s, cp = κR/(κ − 1), κ = 1.4, and R = 287.0J/kg K. The influence of temperature on cp is neglected. ∗ ∗ The parameters defining the problem are η(η , S, T ), R, κ, k, To, po, , W, and g . The independent dimensionless parameters appearing in the problem are , κ, ∗ ∗ Pr, Ra, s/T , To/T , and ρo = po/RTo. The problem is completely defined by the Ra number value, the value of , a reference state po and To, and initial conditions, e.g. T(x, y) = To, p(x, y) = po, and v (x, y) = 0. The computations were performed for Ra = 103, Ra = 104, Ra = 105, and Ra = 3.1 × 105, with the temperature difference parameter = 0.6, a result of imposed Th = 960 K and Tc = 240 K. Figures 9 and 10 show velocity and temperature fields at the steady state.An evident difference against the incompressible flow computations is the asymmetry of the flow and temperature fields.

Figure 9: Velocity ﬁeld and temperature isolines, compressible ﬂow: Ra = 103, 104, left to right.

Figure 10: Velocity ﬁeld and temperature isolines, compressible ﬂow: Ra = 105.

Table 1: Average Nusselt number values for different Rayleigh number values for incompressible and compressible ﬂuid models.

Ra 103 104 105 3.1 × 105

Incompressible 1.056 1.790 3.377 4.477 [18] – – – 4.493 [20] – – – 4.579 Compressible 1.060 1.807 3.321 –

Table 1 shows the comparison of the computed Nusselt number values for different Rayleigh number values for incompressible and compressible ﬂuid models.

10 Conclusions

In the chapter the boundary element integral approach for the solution of compressible fluid motion in a thermally driven cavity is presented. The governing equations arise from the velocity–vorticity formulation of the Navier–Stokes equation for a compressible fluid, and consist of flow kinematics, vorticity transport, heat transport, and pressure equations. The flow kinematics and pressure equations are solved by using a false transient approach in order to increase the stability of the nonlinear iteration procedure. The variable material properties are accounted for in pseudo-body force terms, discretised by internal cells. The presented numerical scheme can further be improved by the use of subdomain-based elliptic difusion– convection fundamental solutions in deriving the integral representations of flow kinetics and energy transport equations. The results of the computed test example of buoyant flow of compressible fluid in a tall cavity confirm the applicability of a BEM-based numerical scheme also for a highly nonlinear transport phenomena such as compressible natural convection. The computed BEM results are in good agreement with the benchmark solutions presented in [17], especially taking into account the much coarser mesh applied for the BEM numerical model.

References

[1] Batchelor, G.K., An Introduction to Fluid Dynamics, Cambridge University Press: Cambridge, 1967. [2] Wrobel, L.C., The Boundary Element Method, Vol. 1, Applications in Thermo-Fluids and Acoustics, Wiley: Chichester, 2002. [3] Škerget, L., Hriberšek, M. & Žuniˇc, Z., Natural convection ﬂows in complex cavities by BEM. Int. J. Num. Meth. Heat & Fluid Flow, 13(6), 2003. [4] Wu, J.C. & Thompson, J.F., Numerical solution of time dependent incompressible Navier-Stokes equations using an integro-differential formulation. Computers and Fluids, 1, pp. 197–215, 1973.

[5] Wu, J.C., Problems of general viscous fluid flow (Chapter 2). Developments in BEM, Vol. 2, Elsevier Appl. Sci. Publ.: London and New York, 1982. [6] Škerget, L. & Samec, N., BEM for the two-dimensional plane compressible fluid dynamics. Eng. Anal. Bound. Elem., 29, pp. 41–57, 2004. [7] Rizk, Y.M.,An integral representation approach for time dependent viscous flow, PhD Thesis, Georgia Institute of Technology, 1980. [8] Škerget, L., Alujeviˇc, A., Brebbia, C.A. & Kuhn, G., Natural and forced convection simulation using the velocity–vorticity approach. Topicsin Boundary Element Research, Vol. 5, pp. 49–86, Springer-Verlag: Berlin, 1989. [9] Škerget, L., Hriberšek, M. & Kuhn, G., Computational fluid dynamics by boundary domain integral method. Int. J. Num. Meth. Eng., 46, pp. 1291– 1311, 1999. [10] Hriberšek, M. & Škerget, L., Iterative methods in solving Navier–Stokes equations by the boundary element method. Int. J. Num. Meth. Eng., 39, pp. 115–139, 1996. [11] Hriberšek, M., Segmentation based boundary domain integral method for the numerical solution of Navier–Stokes equations. BEM 24, Split, WIT Press: Southampton, 2003. [12] Ramšak, M. & Škerget, L., Mixed boundary elements for laminar flows. Int. J. Num. Meth. Fluids, 31, pp. 861–877, 1999. [13] De Vahl Davis, G., Natural convection of air in a square cavity: a bench mark numerical solution. Int. J. Num. Meth. Fluids, 3, pp. 249–264, 1983. [14] Le Quere, P.,Accurate solutions to the square thermally driven cavity at high Rayleigh number. Computers Fluids, 20(1), pp. 29–41, 1991. [15] Jecl, R. & Škerget, L., BEM for natural convection in non-Newtonian fluid saturated square porous cavity. Eng. Anal. Bound. Elem., 27, pp. 963–975, 2003. [16] Vierendeels, J., Merci, B. & Dick, E., Benchmark solutions for the natural convection heat transfer problem in a square cavity. Advances in fluid mechanicsIV, AFM 2002, eds. M. Rahman, R. Verhoeven & C.A. Brebbia, WIT Press: Southampton and Boston, pp. 45–54, 2002. [17] Vierendeels, J., Merci, B. & Dick, E., Numerical study of the natural convection heat transfer with large temperature differences. Int. J. Num. Meth. Heat & Fluid Flow, 11, pp. 329–341, 2001. [18] Ingber, M.S., A vorticity method for the solution of natural convection flows in enclosures. Int. J. Num. Meth. Heat & Fluid Flow, 13, pp. 655– 671, 2003. [19] Weisman, C., Calsyn, L., Dubois, C. & Le Quere, P., Sur la nature de la transition a l’instationare d’un ecoulement de convection naturelle en cavite differentiellement chauffee a grands ecarts de temperature. Comptes rendus de l’academie des sciences, Serie II b, Mecanique, pp. 343–350, 2001. [20] Xin, S. & Le Quere, P., An extended Chebyshev pseudo-spectral contribution to the CPNCFE benchmark. Computational Fluid and Solid Mechanics, Proc. of the First MIT Conf. on Computational Fluid and Solid Mechanics, Vol. 2, Elsevier: Amsterdam, pp. 1509–1513, 1991.