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 restric- tion 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 WIT Transactions on State of the Art in Science and Engineering, Vol 14, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 36 Domain Decomposition Techniques for Boundary Elements 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.