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By Page 7. Reynolds Stress Models 42 ~~ " -. ,. INTRODUCTION TO TURBULENCE MODELLING by Helge I. Andersson Lecture Notes in Subject 76572 Turbulent Flow Division of Applied Mechanics Department of Physics and Mathematics Norwegian Institute of Technology Trondheim, October 1988 TABLE OF CONTENTS page 1. Introduction 1 2. The generalized eddy-viscosity concept 2 3. .rurbulent conductivity 5 4. Zero-equation models 7 4.1 Constant eddy-viscosity models 9 4.2 Mixing-length models 10 4.3 Characterizing features 14 5. One-equation models 17 5.1 Transport equation for k 19 5.2 Transport equation for the eddy viscosity 25 5.3 Transport equation for the shear stress 25 5.4 Characteristic features 27 6. Two-equation models 29 6.1 The k-~ model of turbulence 30 6.2 Numerical examples 33 6. 3 ~_dvantages and disadvantages 40 7. Reynolds stress models 42 .' : References for supplementary reading 46 :,~ i ;; - 1 - 1. INTRODUCTION It is generally accepted that turbulent flows are exactly " represented by three-dimensional time-dependent Navier-Stokes equ- ations, which are the second-order Chapman-Enskog approximation to the Boltzmann equation for molecular motion. Althoug existing computer algorithms arid programs are capable of solving the full Navier-Stokes equations, the storage capacity of present day compu- ters is too small to allow the resolution of the tiniest small- scale fluctuations of the turbulence. Most flows of engineering importance are turbulent, and the desire to make practical calculations useful in design and planning has led to the development of approximate methods which make computer simulations of turbulent flows feasible. The first step towards fairly general prediction methods is the Reynolds-averaging of the governing conservation equations. However, while the process of carefully averaging the momentum and thermal-energy equations can be considered an exact procedure, the resulting averaged equ- ations do not contain enough information about the turbulence to form a soluble set of equations. For example, unknown averages of products of fluctuating quantities: T.. = - P ~ (1) 1J 1 J QJ. = pc u-:--e (2) P J appear in the Reynolds averaged momentum and thermal-energy equation, respectively. Here, eq. (1) represents the apparent turbulent stresses (Reynolds' stresses), while eq. (2) represents an apparent turbulent heat conduction in the j'th direction. It should be remembered, however, that the turbulent stresses and heat flux account for the physical mechanisms of turbulent transport (convec- tion) of momentum and heat, respectively. The process of replacing the unknown averages of products of fluctuating quantities with equations for (or functions of) variables which can be considered as dependent variables in the problem is called modelling. For instance, an equation that suppos- edly expresses a relation between the unknown Reynolds stresses (1) and the mean velocity components is a turbulence model. - - 2 - A turbulence model can be either an algebraic relation or a set of transport equations for some characteristic turbulent quantities. The various models are often classified according to the number of extra partial differential equations which are considered in addition to the Reynolds-averaged Navier-Stokes equ- ations. By selecting a proper turbulence model, the total number of equations involved equals the number of unknown variables, and a closed set of governing equations is obtained. The objective of these lecture notes is to outline the essen- tial ingredients involved in modern turbulence modelling, and to demonstrate the effectiveness and capabilities of some representative models at each level (according to the above classification scheme). We are focussing on the various models required for the solution of the Reynolds averaged momentum equation, from which the basic ideas and principles are illuminated. The modelling required for the solution of the thermal-energy equation follows basically the same philosophy and strategy. Thus, only a brief account on the modelling of the turbulent heat flux is given in section 3. 2. THE GENERALIZED EDDY-VISCOSITY CONCEPT The convenience of the eddy-viscosity concept is coupled with the mathematical convenience of retaining the same form of the governing equations for laminar and turbulent flow, and there- by allowing for the use of the same solution procedure in both cases. Considering, for instance, the free shear flows like the turbulent jet, wake and mixing-layer, the averaged streamwise moment- um equation become: i .. i " u ~ + V ~ = - a~ (uv) (3) ~, The orginal idea of Boussinesq was to define a turbulent or eddy- viscosity by the relation - PUV- = pvT ayau (4) :r;:,~;;;i;:;;i~*;,?~:i; - 3 - If we substitute eq. (4) into eq. (3), the momentum equation become' u axau + V ayau =aya (vT -ayau ) (5) The eddy-viscosity defined by eq. (4) is not a physical property like the molecular viscosity, but varies with local flow condi- tions and the geometry of the problem. If the eddy-viscosity is assumed to be constant, as suggested by Boussinesq in 1877, equ- ation (5) becomes mathematically equivalent with the streamwise momentum equations for the corresponding laminar free shear flows. The value of the eddy viscosity, which can be obtained from experi- ments, will generally be significantly greater than the molecular viscosity. For the free shear flows, only one of the Reynolds stresses (1) appear in the momentum equation (3). In the general case, however, relations like eq. (4) are required for all nine compo- nents of the Reynolds stresses. In analogy with the molecular stresses cri j = 2 1.1 5 i j ( 6) the turbulent stresses can be modelled according to 1.' T" = - p~ = 2pv 5 )' - - 1.' )' it.". (7) 1.) 1. ) -r 32pko ,-~, ~~'\ Here, the symbol k denotes the~kinetic energy of the turbulent motion, defined as k =: , ~ (8) 1. 1. which is a measure of the normal Reynolds stresses. The latter term in the model (7) has been included to assure that the iden- tity : - 4 - . ." - 2 ko = - 2k , - uiui = 2\),Sii -"3 ii ' (9) is recovered for i=j. An important feature of eq. (7) is that the turbulent shear stress ( aU av - p -uv = P\)! ay + ~ ) (10) reduces to the simpler relation (4) for shear flows in which the dominating velocity gradient is the cross-stream gradient of the streamwise velocity component u. Thus, eq. (7) represents a gen- eralized eddy-viscosity hypothesis. Now, if we introduce the eddy-viscosity defined by eq. (7) into the Reynolds averaged momentum equations -aui + u -aui = -1 - a T (11) at j ax. pax. ij ) ) where T.. = - Po.. + 2~S.. - p u-:-i::i-:- 1.) 1.). 1.) 1. ) (12) is the total average stress tensor, the resulting equations become: au. aui a + [ p + 2 ] a [ 1 -.:!::.at u.J -ax. = - ox.'\- -Pox. _3 k + '\- 2 (\) + \) ,1.J)5.. J (13) ) 1. J - 5 - " These fundamental equations reveal~ that the effective viscosity is the sum of the molecular and turbulent (eddy) viscosities. Fur-' thermore, the turbulent kinetic energy k is the turbulent equivalent to the averaged static pressure. Thus, if the kinetic energy is not treated as a dependent variable of the model equations, k is conveniently absorbed in the unknown kinematic pressure. 3. TURBULENT CONDUCTIVITY Following the idea behind the eddy viscosity concept, the turbulent heat flux (2) is related to the gradient of the mean temperature field according to ae - pcp -Uj6 = KT"a"'Xj (14) where Kt is the turbulent conductivity (or, alternatively, the eddy conductivity). This relation is analogous to Fourier's law for heat flux due to molecular agitation. Now, if the assumption (14) is introduced in the averaged thermal energy equation [ ae ae ) - - - 2.9- Pc - + U. -;;-- - ~<t> ax. (15) p at J OXj J where ae - Q : - K- + pc u.e (16) ax. p J J denotes the total mean heat flux, the resulting equation becomes ~-~- ~-~ ~~ . ..~ , - 6 - ! \ . j [ ae ae ] - a pcp a-t + U j ax; = ~<I> - axj [ - (\ K + ~ ) ~ae ] (17) This equation demonstrates that the turbulent conductivity adds to the molecular conductivity to enhance the total heat fluxes. In analogy with the thermal diffusivity caused by molecular action, it may often be convenient to introduce the turbulent or eddy diffusivity: YT :: KT/pcp (18) for the turbulent heat transfer. It should be emphasized, however, that the turbulent diffusivity as well as the turbulent conductivity are not fluid properties but depend on the mean flow and the turbu- lence. These scalar quantities may therefore vary significantly from point to point within a flow field. The eddy-diffusivity hypothesis (14) is analogous to the eddy-viscosity hypothesis (4), i.e. the turbulent transport of a certain property is related to the gradient of that property. Thus, eq. (14) is based on the assumption that the turbulent tran- sport of heat is provided by similar processes as the turbulent transport of momentum. The ratio of the respective coefficients of diffusion is defined as the turbulent Prandtl number: '0" = \) /Y (19) T T T In spite of the fact that the eddy viscosity and the eddy diffu- sivity may vary considerably throughout a flow field, it is experi- mentally verified that the ratio (19) is approximately constant within a flow. Furthermore, the maguitude of the turbulent Prandtl number is approximately constant for a wide range a flow situations. 1 - 7 - - This is the famous Reynolds analogy from 1874. Today, engi- . nee ring heat and fluid flow calculations take advantage of this analogy and take the turbulent Prandtl number to be a constant close to 1. 4. ZERO-EQUATION MODELS Turbulence models using only partial differential equations for the mean velocity field, and no differential equation for the turbulence are classified as zero-equation models. All models belonging to this class use the generalized eddy-viscosity concept (7). The eddy-viscosity is furthermore related to the mean flow field via an algebraic relation. Therefore, these models are also called algebraic models.
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