DEVELOPMENT OF A CARTESIAN GRID BASED CFD SOLVER (CARBS) by A.M. Vaidya, N.K. Maheshwari and P.K. Vijayan Reactor Engineering Division

Transcription

1 BARC/2013/E/023 BARC/2013/E/023 DEVELOMENT O A CARTESIAN GRID BASED CD SOLVER (CARBS) b A.M. Vaida, N.K. Maheshari and.k. Vijaan Reactor Engineering Division

2 BARC/2013/E/023 GOVERNMENT O INDIA ATOMIC ENERGY COMMISSION BARC/2013/E/023 DEVELOMENT O A CARTESIAN GRID BASED CD SOLVER (CARBS) b A.M. Vaida, N.K. Maheshari and.k. Vijaan Reactor Engineering Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2013

3 BARC/2013/E/023 BIBLIOGRAHIC DESCRITION SHEET OR TECHNICAL REORT (as per IS : ) 01 Securit classification : Unclassified 02 Distribution : Eternal 03 Report status : Ne 04 Series : BARC Eternal 05 Report tpe : Technical Report 06 Report No. : BARC/2013/E/ art No. or Volume No. : 08 Contract No. : 10 Title and subtitle : Development of a Cartesian grid based CD solver (CARBS) 11 Collation : 73 p., 52 figs., 3 tabs. 13 roject No. : 20 ersonal author(s) : A.M. Vaida; N.K. Maheshari;.K. Vijaan 21 Affiliation of author(s) : Reactor Engineering Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate author(s) : Bhabha Atomic Research Centre, Mumbai Originating unit : Reactor Engineering Division, Bhabha Atomic Research Centre, Mumbai 24 Sponsor(s) Name : Department of Atomic Energ Tpe : Government Contd...

4 BARC/2013/E/ Date of submission : November ublication/issue date : December ublisher/distributor : Head, Scientific Information Resource Division, Bhabha Atomic Research Centre, Mumbai 42 orm of distribution : Hard cop 50 Language of tet : English 51 Language of summar : English, Hindi 52 No. of references : 21 refs. 53 Gives data on : 60 Abstract : ormulation for 3D transient incompressible CD solver is developed. The solution of variable propert, laminar/turbulent, stead/unstead, single/multi specie, incompressible ith heat transfer in comple geometr ill be obtained. The formulation can handle a flo sstem in hich an number of arbitraril shaped solid and fluid regions are present. The solver is based on the use of Cartesian grids. A method is proposed to handle comple shaped objects and boundaries on Cartesian grids. Implementation of multi-material, different tpes of boundar conditions, thermo phsical properties is also considered. The proposed method is validated b solving to test cases. 1 st test case is that of lid driven flo in inclined cavit. 2 nd test case is the flo over clinder. The 1 st test case involved stead internal flo subjected to WALL boundaries. The 2 nd test case involved unstead eternal flo subjected to INLET, OUTLET and REE-SLI boundar tpes. In both the test cases, non-orthogonal geometr as involved. It as found that, under such a ide conditions, the Cartesian grid based code as found to give results hich ere matching ell ith benchmark data. Convergence characteristics are ecellent. In all cases, the mass residue as converged to 1E-8. Based on this, development of 3D general purpose code based on the proposed approach can be taken up. 70 Keords/Descriptors : HEAT TRANSER; RANDTL NUMBER; TURBULENT LOW; NAVIER-STOKES EQUATIONS; VISCOSITY; MESH GENERATION; LUID MECHANICS; COMUTERIZED SIMULATION 71 INIS Subject Categor : S42 99 Supplementar elements :

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6 ABSTRACT ormulation for 3D transient incompressible CD solver is developed. The solution of variable propert, laminar/turbulent, stead/unstead, single/multi specie, incompressible ith heat transfer in comple geometr ill be obtained. The formulation can handle a flo sstem in hich an number of arbitraril shaped solid and fluid regions is present. The solver is based on the use of Cartesian grids. A method is proposed to handle comple shaped objects and boundaries on Cartesian grids. Implementation of multi-material, different tpes of boundar conditions, thermo phsical properties is also considered. The proposed method is validated b solving to test cases. 1 st test case is that of lid driven flo in inclined cavit. 2 nd test case is the flo over clinder. The 1 st test case involved stead internal flo subjected to WALL boundaries. The 2 nd test case involved unstead eternal flo subjected to INLET, OUTLET and REE-SLI boundar tpes. In both the test cases, non-orthogonal geometr as involved. It as found that, under such a ide conditions, the Cartesian grid based code as found to give results hich ere matching ell ith benchmark data. Convergence characteristics are ecellent. In all cases, the mass residue as converged to 1E-8. Based on this, development of 3D general purpose code based on the proposed approach can be taken up. iv

7 Table of Contents List of igures... viii List of Tables... Nomenclature... i 1.0 Introduction Solution rocedure Governing Equations Laminar los Turbulent los Averaging procedure Boussinesq edd viscosit hpothesis Renolds averaged equations Turbulence Modeling Turbulent viscosit Standard k-є turbulence model RNG k-є turbulence model Heat Transfer Modeling Choice of dependent variable in energ equation Thermal Energ Equation Relation beteen temperature and internal energ Solution of Governing Equations MATERIAL ROERTIES Multi-Material Modelling Variable thermo-phsical properties Densit variation in computational domain Densit in Solid Region Densit in luid Region ace centre densities Molecular Viscosit Viscosit in Solid Region Viscosit in luid Region Specific heat at constant volume v

8 5.6 Thermal conductivit INITIAL AND BOUNDARY CONDITIONS Initial conditions Boundar conditions Inlet Outlet of fied pressure tpe Outlet of full developed flo tpe Wall boundar Smmetr boundar Discussion on boundaries DISCRETIZATION Domain Discretiation Equation Discretiation Discretiation method Control volume (or finite volume cell) Scalar transport equation Iterative procedure Time step Grid fineness LINEAR EQUATION SOLVER Gauss Siedel Solver STAGGERED ARRANGEMENT O VARIABLES EQUATION OR RESSURE ALGORITHM O CD SOLVER Semi-Implicit Method for ressure Linked Equations Convergence Monitoring COMLEX GEOMETRY OST ROCESSING Unstead Results Stead Results Test Cases st Test Case : Inclined Lid Driven Cavit Mesh vi

9 15.2 Validation Comparison of CARBS results ith Benchmark data at Re 100 and θ45 o Comparison of CARBS results ith Benchmark data at Re 1000 and θ30 o Convergence characteristics of CARBS Effect of Inclinations on lo attern in Cavit Summar and Conclusions for 1 st Test Case nd Test Case lo over Circular/Square Clinder resent Work Simulation Inputs Mesh Mathematical ormulation Code Development Mesh Independence Time Step Independence Lo Renolds Number Results Circular clinder results Square clinder results Vorte Shedding at High Renolds Number Comparison of erformance of GS and ADI Solvers Summar and Conclusions Bibliograph vii

10 List of igures ig. 1 : Concept of overlapping materials... 9 ig. 2 : Wall boundar and adjacent computational cell ig. 3 : Treatment of internal inlet ig. 4 : A 3D uniform Cartesian grid discretiing a domain of 25 m25 m 25m sie ig. 5 : 3D Control volume and location of cell centre, north interface n, south interface s, east interface e, est interface ig. 6 : Control volume, centre point and neighbours in XY and YZ planes ig. 7 : Staggered arrangement of u and v velocit components ith respect to pressure (in XY plane) ig. 8 : Representation of a circle on a Cartesian grid ig. 9 : Schematic diagram of inclined lid driven cavit ig. 10 : Geometr and mesh ig. 11 : lo pattern computed using CARBS (Re100, θ 45 o ) ig. 12 : U velocit profile along vertical center line for Re100, θ45 o ig. 13 : V velocit profile along horiontal center line for Re100, θ45 o ig. 14 : Re 1000, inclination 30 o ig. 15 : U velocit profile along vertical center line for Re1000, θ 30 o ig. 16 : V velocit profile along horiontal center line for Re1000, θ30 o ig. 17 : Convergence (Re2000, inclination 15 o ) ig. 18 : Streamlines (Re2000, inclination - 90 o ) ig. 19 : Streamlines (Re2000, inclination - 75 o ) ig. 20 : Streamlines (Re2000, inclination - 60 o ) ig. 21 : Streamlines (Re2000, inclination - 45 o ) ig. 22 : Streamlines (Re2000, inclination - 30 o ) ig. 23 : Streamlines (Re2000, inclination - 15 o ) ig. 24 : Schematic diagram of flo over a clinder (NOT TO SCALE) ig. 25 : Non-uniform mesh generation for the domain ig. 26 : Enlarged vie of mesh around clinder ig. 27 : Variation of u component of velocit along horiontal central line of the domain as computed on different grids ig. 28 : Variation of v component of velocit along horiontal central line of the domain as computed on different grids ig. 29 : Variation of static (gauge) pressure along horiontal central line of the domain as computed on different grids ig. 30 : u velocit at center of outlet; as computed b different time steps ig. 31 : Static gauge pressure variation at center of outlet; as computed b different time steps ig. 32 : Convergence histor (Circular clinder at Re40) ig. 33 : lo field around circular clinder at Re ig. 34 : lo field around circular clinder at Re ig. 35 : lo field around circular clinder at Re ig. 36 : lo field around circular clinder at Re ig. 37 : Vorticit field around circular clinder at Re ig. 38 : ressure and velocit fields (Circular clinder Re 40) ig. 39 : lo field around square clinder at Re viii

11 ig. 40 : lo field around square clinder at Re ig. 41 : lo field around square clinder at Re ig. 42 : lo field around square clinder at Re ig. 43 : lo field around square clinder at Re ig. 44 : ressure and velocit field around square clinder at Re ig. 45 : lo field around square clinder at Re ig. 46 : Convergence histor for Re 62.5 (stead state mode) ig. 47 : streamlines at different times (square clinder at Re 100) ig. 48 : variation of lift force ith time ig. 49 : Variation of Strouhl number ith inlet Renolds number for square clinder ig. 50 : Variation of Strouhl number ith inlet Renolds number for circular clinder ig. 51 : Convergence b GS and ADI schemes in initial 1000 iterations ig. 52 : Convergence b GS and ADI schemes in later time period i

12 List of Tables Table 1 : Choice of governing equations... 2 Table 2 : Model constants in standard and RNG k-epsilon model... 6 Table 3 : Default values at inlet boundar... 12

13 Nomenclature c1, c2 turbulence model constants c v g, g, g i specific heat at constant volume, J/kg-K components of gravitational acceleration in, and coordinate directions respectivel, m/s 2 specific internal energ, J/kg k turbulent kinetic energ, m 2 /s 2 l length scale, m mf species mass fraction p static pressure, ascal r T turbulent randtl number q heat flu, W/m 2 TI turbulent intensit u, v, velocit components in, and directions, m/s,, Cartesian coordinates Greek Smbols densit, kg/m 3 T molecular viscosit, a-s turbulent viscosit, a-s change σ k,σ ε constants used in k-є turbulence model Superscripts o scalar variable old time step value Subscripts e,, n, s, f, b east, est, north, south, front and back interface locations respectivel ref reference Abbreviations CD Computational luid Dnamics ORTRAN ormula Translating Language RANS Renolds averaged Navier-Stokes equations RNG Re-normaliation Group Theor i

14 1.0 Introduction CD simulation of industrial sstems is ver comple due to large number of components involved. urther, the geometr of each component ma be comple, non-orthogonal. In such a case, the use of multi-block bod-fitted grids becomes ver cumbersome due to the need to generate a ver large number of blocks. Also, if the geometr is highl skeed, the bod fitted grid becomes highl skeed hich results in poor accurac. Use of unstructured grids based CD formulation is being popular. Hoever, such a simulation needs availabilit of a sophisticated commercial CD mesh generator and solver, hich is ver epensive. urther, it is ver difficult to implement higher order convection schemes in unstructured grids. A simple approach is to use cartesian grids and resolve all components over a single grid. At curved boundaries, some approimation is needed. A smooth curved boundar is modelled as a staircase tpe boundar. Hoever, b using fine grid in regions of curved boundaries, the error beteen actual curved boundar and modelled boundar can be made negligible. Use of cartesian based approach offers folloing advantages over bod fitted and unstructured grids. A Structured code can be developed. Efficient solution methods for structured grids are available. It is possible to easil implement staggered arrangement of pressure and velocit, hich is ver robust and handles non-linear pressure profiles ith ease. There is no need of generating multiple blocks. A single grid spans entire domain. Grid generation is ver efficient. There is no need to solve an additional partial differential equation to generate grid. Higher order schemes like QUICK, 2 nd order upind, etc, can be implemented. Governing equations in cartesian forms are readil available. Even if the sstem under consideration involves high non-orthogonalit, the grid orthogonalit is not affected. Thus, there is no effect of sstem geometr on convergence behavior of the code. A formulation is developed hich is 3D and transient. A test program (2D) is developed based on the formulation. The test cases involve laminar/turbulent flos, comple geometr, eternal as ell as internal flos, stead and unsteadiness. Thus, the testing under idel varing conditions is performed and ver encouraging results are obtained. The report contains the formulation and results of test cases. 1

15 2 2.0 Solution rocedure irst governing equations are identified based on phsics of the flo. These are partial differential equations (describing conservation of mass, momentum, etc.). Using suitable discretiation scheme, e.g. finite volume, these equations are discretied i.e. converted into algebraic equations. The computational domain is also discretied (in this approach, using cartesian grids!). Then, subject to (i) specified boundar and initial conditions, (ii) material properties, (iii) sources and (iv) sstem geometr, these conservation equations are solved. These governing equations, boundar conditions, material properties calculation and solution procedures are eplained in this report. 3.0 Governing Equations Choice of governing equations depends on phsics of flo. These are summaried in Table 1. Table 1 : Choice of governing equations Laminar flos Navier Stokes equations Turbulent flos Renolds averaged Navier Stokes equations along ith equations representing turbulence model Conduction Heat Transfer Energ equation Simultaneous lo and Heat Transfer Momentum and energ equations Multi-component flos Species conservation equation These equations are revieed in this section. 3.1 Laminar los 3D transient laminar flos can be computed b solving momentum equations in, and directions. These equations, in strong conservation form, are given belo (Bird et al. 3 ). X direction momentum equation for laminar flo is given belo. g v v 3 2 p u u u vu u 3 4 uu t u (1) Y direction momentum equation for laminar flo is given belo. g u u 3 2 p v v v 3 4 vv v uv t v (2) Z direction momentum equation for laminar flo is given belo.

16 3 g v u v u 3 2 p 3 4 v u t (3) These equations are coupled. In segregated iterative manner solution of these equations can be obtained. Additionall, solution of these equations requires computation of pressure field. This ill be discussed later. These equations are applicable to completel variable propert flo. 3.2 Turbulent los or modeling turbulent flos, the RANS (Renolds Averaged Navier Stokes) equations are used (Launder and Spalding 14 ). To obtain RANS from above specified laminar equations, folloing mathematical treatment is carried out Averaging procedure 1. Split scalars into mean and fluctuating component and rite the epanded form of equation. ( ( ) ( ) ( ) t t t,,, ) 2. Time average each term of the equation. ( ) ( ) ( ) dt t t dt t t dt t t, 1, 1, 1 here ( ) ( ) t dt t t,, 1 ; ( ) 0, 1 dt t t 3. Use Boussinesq edd viscosit hpothesis (Eq. 4) for modeling the turbulent stresses Boussinesq edd viscosit hpothesis ij i i T i j j i T j i u k u u u u δ 3 2 ' ' (4) The momentum equations for turbulent flos, thus obtained, are given belo Renolds averaged equations U - momentum equation for turbulent flo is as follos. ( ) ( ) ( ) ( ) ( ) k g v v p u u u vu u uu t u T T T T T (5) V - momentum equation for turbulent flo is as follos.

17 4 ( ) ( ) ( ) ( ) ( ) k g u u p v v v vv v uv t v T T T T T (6) W - momentum equation for turbulent flo is as follos. ( ) ( ) ( ) ( ) ( ) k g v u v u p v u t T T T T T (7) It can be noted that the equations for laminar flos (i.e. Eq. 1,2,3) can be obtained from equations for turbulent flos (i.e. Eq. 5,6,7) b just setting T and k to ero. This helps in developing same code hich serves for laminar as ell as turbulent flos. 3.3 Turbulence Modeling Turbulent viscosit The Renolds averaged momentum equations given above contain turbulent viscosit ( T ) hich needs to be obtained from a turbulence model. The turbulent viscosit field is obtained from folloing epression. ε 2 k T c (8) In above equation, c is a constant and its value depends on the turbulence model chosen. Hence turbulent viscosit depends on turbulent kinetic energ and its dissipation rate. To obtain these to parameters, to equations are solved. These equations constitute the turbulence model. Thus, turbulent viscosit becomes the link beteen turbulence model and RANS equations Standard k-є turbulence model This model solves equation for turbulent kinetic energ and є each (Launder and Spalding 14 ). Equation for turbulent kinetic energ (3D unstead form) is given belo. ( ) ε σ σ σ G k k k vk k uk t k k T k T k T (9) In above equation, k σ is generall taken to be unit and G is the rate of generation of tke. G term is modeled as follos.

18 5 i j j i j i T u u u G (10) Thus, the rate of generation of turbulent kinetic energ is considered to be directl proportional to velocit gradients. Epanding the above epression in 3D, b putting i1,2,3 and j1,2,3, and after some mathematical manipulation, one gets folloing form v u v u v u G T (11) 3D unstead form of equation for rate of dissipation of turbulent kinetic energ is given belo. k c G k c v u t T T T ε ε ε σ ε ε σ ε ε σ ε ε ε ε ε (12) In above equation, C11.44, c21.92 and ε σ 1.3 are generall accepted values. The value of G required in above equation is obtained from Eq. (10) itself. After obtaining the tke and є fields, the turbulent viscosit field is obtained from Eq. (8). Value of c required in Eq. (8) is 0.09 in standard k- є model RNG k-є turbulence model Turbulence model has significant effect on accurac of predictions. In vie of this, advanced turbulence modeling should be available in general purpose softare. Hence the RNG K- є turbulence model is also proposed to be incorporated in this CD solver. In flos involving acceleration and large strain rate, standard k-є turbulence model is knon to over predict effect of turbulence on transport of momentum, energ, etc. RNG k- є turbulence model is established model hich offers better accurac in accelerating flos. It is also better suited for flos involving streamline curvature. The difference beteen standard k- є model and RNG k- є turbulence model is in the є equation and also the model constants. Equation for tke is same as than in standard k- є model (Eq. 9). But, in standard k-є model, k σ is taken to be unit, hereas, in RNG k-є model, it is taken to be

19 Equation for є is given belo. 2 ε T ε T ε T ε ε ε uε vε ε c1 G c2 R (13) t σε σε σε k k In standard k-є model, equation for є does not have R term on RHS. This terms forms main difference beteen standard and RNG model versions. The model constants are also different and this is summaried in Table 2.. Table 2 : Model constants in standard and RNG k-epsilon model Constant Standard k-є RNG k-є c σ k σ ε c c The modeling of R term in Eq. (13) is done as follos. 3 η c η 1 2 η 0 ε R 3 ( 1 βη ) k here η , β here S ( 2S S ij ij ), k and η S, ε (14) here S ij 1 u 2 i j u j i It can be noted that, for a rapidl accelerating flo, the value of η increases and once ( 1η η0 ) < 0, the value of R becomes negative. This increases the value of ε and thus reduces the value of tke in a cell. Thus the prediction of transport rates using RNG model ill be more accurate and this model avoids the over prediction of tke in a rapidl accelerating flo. 3.4 Heat Transfer Modeling Heat transfer is an important aspect of CD analsis (and especiall for reactor applications). The heat transfer ma involve folloing: 1. Temperature distribution depending on flo rates, heat generation and thermophsical properties, etc. 6

20 2. Heat transfer to/from ambient through convective or isothermal or heat flu boundar condition. 3. orced or natural convection heat transfer in laminar/turbulent conditions. 4. Conjugate heat transfer heat generation in solid and transfer to surrounding liquid or vice versa. The addition of heat transfer modeling b adding thermal energ equation greatl enhances the practical utilit of CD softare Choice of dependent variable in energ equation Thermal energ equation can be derived using either temperature or internal energ or enthalp as the dependent variable. Since a general purpose solver is supposed to solve for solid regions as ell as fluid regions, the enthalp based form ma not be good. Enthalp involves pressure hich is irrelevant for solids. If temperature based form is used, in full variable propert case, during Renolds averaging, one faces difficulties and onl an approimate form (hich ill reduce the accurac of computed temperature field) can be derived. Internal energ based form does not suffer from these limitations and hence is used in the softare Thermal Energ Equation Vector form of thermal energ equation is given belo (Bird et al. 3 ). () q p( V ) V i t (15) Laminar form Above equation is epanded to give folloing 3D unstead form in Cartesian coordinates for LAMINAR flos ith full variable properties. i ui t Turbulent form k C v i vi k C v i i k C v i u v q p or TURBULENT flo, the internal energ and velocities are decomposed into mean and fluctuating components and time averaging is performed. The resultant stress terms are modeled as per folloing gradient diffusion hpothesis (Launder and Spalding 14 ). / T ui' σ i (16) (17) 7

21 And finall e get the folloing form. i k ui t Cv T r T i vi k C v T r T i i k C T r u v q p v T i (18) Tpicall the value of r T is The above form is suitable for modeling pure conduction case also Relation beteen temperature and internal energ Generall the phsical boundar conditions are knon in terms of temperature. If the internal energ based form is used, it ill be necessar to convert temperature based conditions into internal energ based conditions and then solve Eq. 18 to get internal energ field and then ould eventuall convert i field into T field. The inter-conversion beteen I and T fields depends on hether the specific heat is constant or variable. Constant ropert case Relation beteen internal energ and temperature is given belo. i i ref C v ( T T ) ref The equation can be used to convert i into T or T into i, as required. Variable propert flo modelled using polnomials In this case, to relations ill be required. ( T ) and T T ( i) i i These ma be polnomials of an order. The user ill specif the order and constants of these polnomials. Variable propert flo modelled using tabulated propert data The user ill specif the internal energ dependence on temperature in a tabulated manner and the inter conversion of i and T fields ill be done b solver using piece-ise linear interpolation performed on the tabulated data. 4.0 Solution of Governing Equations or solving the governing equations mentioned in Section 3.0, folloing considerations appl. The solution depends on material properties (i.e.,, cp and k). The equations are subjected to initial and boundar conditions. The equations are partial differential equations and hence need to be discretied to obtain corresponding algebraic equations. (19) 8

22 Momentum equations contain pressure gradient term. Hence an equation for pressure ould be required. Equations are dependent on each other. A suitable procedure/algorithm is needed to solve the coupled set of equations. Solution of simultaneous set of equations needs to be performed in efficient manner. These issues are discussed in forthcoming sections. 5.0 MATERIAL ROERTIES 5.1 Multi-Material Modelling To be able to handle industrial problems, it is required to handle arbitraril an number of materials in the computational domain. The concept of overlapping materials is useful for this. This concept is eplained ith the help of ig. 1. If object 1 is inserted first and then 2 nd, then the material in one A ill be that of object 1 and material in ones B and C ill be that of 2 nd object. If the 2 nd object is inserted first and then object 1, in that case, one A and one B ill be object 1 material and one C ill be filled ith object 2 material. Hoever, it must be understood that, in each individual finite volume cell, onl 1 material ill be filled. A computational cell contains onl one material either fluid or solid (depending on material). ig. 1 : Concept of overlapping materials 5.2 Variable thermo-phsical properties The spatial distribution of densit and viscosit is required to solve the momentum equations. urther, for modeling heat transfer, apart from densit, specific heat and thermal conductivit are also required. 9

23 The thermo-phsical properties are dependent on temperature and pressure. But incompressible flo solver is supposed to solve flo problems in hich operating pressure remains constant. Hence the thermo-phsical properties are dependent on temperature onl. This is discussed net. 5.3 Densit variation in computational domain In a computational domain in hich there are arbitraril an number of solid and fluid regions, densit varies throughout the domain. Densit is evaluated at cell centre i.e. point shon in ig Densit in Solid Region If a cell is in solid region, its densit is constant. CD solver precludes accounting for selling of solid objects ith rise in temperature. Hence densit of cells hich are in solid region is constant and is equal to user specified value Densit in luid Region If a cell lies in fluid region, then its densit is decided from folloing considerations Single component flo Densit can be constant (equal to user specified value at reference temperature), or, it ma be function of temperature. Such a dependence on temperature ma be specified in the form of polnomial or in the form of tabulated data. olnomial representation of densit In this case, the densit is epressed as follos. a b T C T 2... (20) The temperature is in Kelvin. Tabulated representation of densit The tabulated data consisting of temperature (in Kelvin) in 1 st column and densit (in kg/m 3 ) in 2 nd column must be specified in a data file and the name of the data file has to be specified to the softare Multi-component flo In a multi-component flo, densit ould be computed from component mass fractions. Thus, N i 1 mf i i (21) here i component densit and mf i is mass fraction of i th specie and N total number of species. i ma be constant or ma be dependent of temperature (and represented b polnomial or tabulated propert data). 10

24 5.3.3 ace centre densities The previous discussion is about densit at cell centre. The densit at cell face centres ill be required for computing the mass flues. The densit at cell face centres ill be evaluated b linear interpolation using the cell centre values. 5.4 Molecular Viscosit Viscosit in Solid Region Viscosit in solid region ould be set to a ver large value so as to reduce the velocities to ero. The use of harmonic mean for computing face centre values ill be mandator ith this approach Viscosit in luid Region or single component flo or a specie/component of a multi-component flo, the viscosit ma be set to constant or ma be dependent on temperature. The temperature dependence ma be epressed as a polnomial or tabulated data. The user ill then either specif the order and constants of polnomial or specif the name of data file in hich tabulated data of Viscosit vs. temperature is specified. In a multi-component flo, the viscosit of each species is computed from above and then miture viscosit is computed using folloing epression. N i 1 mf i i 5.5 Specific heat at constant volume Since the internal energ based thermal energ equation is being solved, hence the user ill have to specif specific heat at constant volume, c v, (for each material involved in simulation). The specific heat ma be constant or variable (function of temperature). In later case, it ma be specified in the form of polnomial or in the form of tabulated data. In case of polnomial, the user ill specif the order of polnomial and the values of constants. In case of tabulated data, the user ill create an ASCII file in hich first column contains temperature (K) and 2 nd column contains c v. The user ill then specif the name of this file to the softare. 5.6 Thermal conductivit Thermal conductivit is required to be specified if heat transfer is being modelled. Again, constant or variable thermal conductivit ma be specified in the form of polnomial or tabulated data. (22) 11

25 6.0 INITIAL AND BOUNDARY CONDITIONS 6.1 Initial conditions The implementation of eact initial conditions is required in unstead flos. In stead cases, the initial condition acts as initial guess. During restart, initial conditions are same as conditions computed in last iteration/time step of previous run. In an case, the implementation of initial condition is straightforard. 6.2 Boundar conditions The specific solutions of the governing equations can be obtained b subjecting them to boundar conditions. The folloing boundar conditions ill be available in the softare. Inlet Inlet Outlet of fied pressure tpe Outlet of full developed flo tpe Wall Smmetr At inlet boundar, the three velocit components, temperature, tke and є ill be specified b the user, depending on the models activated. Default values are specified in Table 3. The turbulent kinetic energ ill be computed from turbulent intensit and velocit using folloing epression (Bisas and Esaran 4 ). k inlet 3 2 ( U TI ) 2 inlet Rate of dissipation of turbulent kinetic energ at inlet ill then be computed from folloing epression. ε inlet c kinlet l It ma be noted that the inlet ma be at the actual boundar of the computational domain or it ma be internal inlet as ell. Arbitraril an number of outlets can be created. Table 3 : Default values at inlet boundar arameter Default value Velocit component Temperature 0.0 m/s K Turbulent intensit 10% Length scale Equal to sie of inlet (23) (24) 12

26 6.2.2 Outlet of fied pressure tpe The user need specif the gauge pressure at the outlet. Default value is ero gauge pressure. Since the solver solves for in-compressible flo, hence it deals ith gauge pressure. There is no need to bring absolute pressure into computations. The absolute pressure ill be available at post-processing session. The use of gauge pressure reduces round-off error associated ith computing pressure gradients in momentum equations. There ma be an number of outlets. The outlets ma be at actual boundar of the computational domain or ma be internal as ell. The use ma specif different pressures at different outlets if desired Outlet of full developed flo tpe In this boundar tpe, the gradients of velocit/tke/є/temperature normal to the boundar are equated to ero. The pressure need not be specified. It ill be adjusted in such a a that the resulting velocities ill satisf mass balance over the cell. These boundaries should be located at or close to boundar of the computational domain. The should not be located deep inside the domain far aa from boundaries Wall boundar A all boundar offers no-slip boundar for velocit component hich is tangential to it and impermeable boundar for velocit component normal to it. The all is treated as ero thickness object. The user ma be alloed to insert a all at domain boundar or internal to it. Arbitraril an number of alls ma be inserted. The all ma be stationar or moving. Hoever, the movement is alloed in directions hich are tangent to the orientation of all. That is, if a all is kept in YZ plane, then it is NOT alloed to have component of velocit but it can have Y and Z components of velocit. This makes it sure that the all doesn t compress the domain ahead or behind it. In case of moving all, phsicall the all is not moved but the effect of motion of a all on adjacent fluid is accounted for using no-slip boundar condition. The mathematical treatment internall performed b the softare need not be of concern to the user but is specified briefl here. In case of laminar flo, the no-slip boundar condition is used to estimate the all shear stress in the cell hich is adjacent to the all (as shon in ig. 2). This all shear stress provides required resistance for all adjacent tangential velocit component. 13

27 In case of turbulent flos, the all shear stress is not computed from no-slip boundar condition. Rather, it is estimated from standard all functions hich are based on logarithmic velocit profiles (Launder and Spalding 14 ). ig. 2 : Wall boundar and adjacent computational cell Smmetr boundar Smmetr boundar condition is ver useful in simulating comple but smmetric problems. In comple problems, fine mesh is required. Use of smmetr boundar helps to reduce the etent of computational domain b half or quarter thus enabling reasonable number of cells. The other use of smmetr boundar is the abilit to perform 2D simulations ith the 3D solver. Consider 2D simulation needs to be performed, then user ill create the geometr in XY plane and in direction, he/she ill set a single cell and ill place a smmetr boundar on Z 0 and Z Zma planes. In this case, in direction, computational domain thickness can be an arbitrar value like 1 m or so. The thickness in -direction ill not affect the simulation results. urther, it is useful to check correctness of solver to be developed. Take a smmetric domain. erform a 3D simulation on full domain. Then perform 3D simulation on half of the domain including smmetric boundar. Both these simulations should give same result. Unlike inlet/all/outlet, the smmetr boundar ill be onl at boundar of the computational domain. Internal smmetric boundaries are not permissible. At smmetr boundar, the normal gradients of all scalars ill be set to ero. lo can t cross a smmetr boundar because in that case it ill spoil smmetr of the flo field. Hence the velocit component normal to a smmetr boundar ill be directl set to ero. 14

28 6.2.6 Discussion on boundaries Inclined boundaries Inserting boundar tpes hich are aligned ith Cartesian aes is straightforard. In case inclined inlet needs to be inserted, then one has to insert an inlet hich is aligned ith Cartesian ais and then adjust the velocit components. Internal boundaries Consider an internal inlet placed at X Constant location. The direction component ma be set to specified value hich ma be positive or negative. The other side ill be treated as a WALL tpe boundar. This is illustrated in ig. 3. Thus ve or ve velocit component decides the inlet side and all side. Such a representation is more realistic of the actual phsical situation in hich an internal inlet represents a nole inserted in the sstem. One side of nole ill act as inlet hereas other side ill act as all. Same philosoph applies for outlets as ell. 7.0 DISCRETIZATION 7.1 Domain Discretiation The fluid mechanics deals ith continuum phsics and the governing equations described earlier represent continuum phsics. But since analtical solution of the equations is not possible, hence the equations need to be solved b numerical techniques. The numerical techniques cannot handle partial differential equations. The can handle onl algebraic equations (Balagurusami 1 ). The conversion of partial differential equations into algebraic equations is called equation discretiation. The discretied equations are solved over discretied computational domain. The domain is discretied b generating a grid. In this process, a continuum field is approimated b a set of discrete values and hence the process is called discretiation. In this solver, the grid is generated b draing Xconstant, Yconstant and Z constant lines throughout the computational domain. The lines ma be uniforml spaced or ma be non-uniforml spaced. A tpical 3D grid is shon in ig Equation Discretiation Discretiation method The conversion of partial differential equations into equivalent algebraic equations is carried out using finite volume method hich is a popular discretiation method (atankar 17 ). The finite volume method offers man advantages: it satisfies conservation of mass, momentum, energ over coarse grid also, - it operates on integrated equation and hence involves 1 st order and ero order derivatives. 15

29 ig. 3 : Treatment of internal inlet ig. 4 : A 3D uniform Cartesian grid discretiing a domain of 25 m25 m 25m sie Due to this, implementation of boundar conditions becomes straight forard. - The integrated equation contains terms hich represent phsicall meaningful quantities like shear stress, heat flu, etc. hence the modeling of these terms in consistent ith the phsics can be performed. 16

30 7.2.2 Control volume (or finite volume cell) The computational domain is discretied b draing interfaces. Heahedral Cells are thus formed. Equations are integrated over these cells. A tpical cell is shon in ig. 5. The integrated convection terms are discretied as per upind philosoph and pressure gradient and diffusion terms are modeled as per central difference method. Transient term is modeled as per backard difference. Source term is algebraic and hence retained in its integrated form. ig. 5 : 3D Control volume and location of cell centre, north interface n, south interface s, east interface e, est interface Scalar transport equation The governing equations described earlier are integrated over a 3D control volume. The form of all the governing equations is same. Each equation contains a transient term, three convection term, three diffusion terms and a source term. Hence the governing equations can be represented b a single equation knon as scalar transport equation. It is given belo. t ( u ) ( v ) ( ) Γ Γ Γ S The scalar represents u, v,, k, є, i, etc., depending on hich equation is under consideration. The equation discretiation is no eplained for the above equation. The equation is integrated over the control volume hose boundaries are -e-s-n-b-f. ig. 6 clearl shos these limits (a 3D cell is shon in XY and YZ planes for better clarit). E and (25) 17

31 18 W denote neighbours in X direction. Similarl, N and S denote neighbours in Y direction and B and denote neighbours in Z direction. ig. 6 : Control volume, centre point and neighbours in XY and YZ planes We get folloing equation. ( ) ( ) ( ) ( ) Γ Γ Γ f b n s e f b n s e f b n s e f b n s e ddd S ddd ddd v u ddd t (26) Let s consider integration of each term one b one. Integration of transient term ( ) ( ) ( ) t ddd t o p f b n s e (27) In this case, the backard difference in time is used to model ( ) t term. It is 1 st order accurate in time. Here, );, ( );, ( );, ( );, ( );, ( );, ( 2 ; 2 ; 2 B d d S d N d W d E d here b f s n e b f s n e δ δ δ δ δ δ δ δ δ δ δ δ Integration of convection terms ( ) ( ) ( ) b b f f s s n n e e f b n s e ddd v u

32 19 here, ; ; ; ; ; ; v v u u b b b f f f s s s n n n e e e These are convective flues. At this point, one needs to relate interface value of scalar ith the cell centre value. or this purpose, 1 st order upind scheme is used. Hence e get, ( ) ( ),0 0, e E e p e e MAX MAX ( ) ( ),0 0, W MAX MAX ( ) ( ),0 0, n N n p n n MAX MAX ( ) ( ),0 0, s s S s s MAX MAX ( ) ( ),0 0, f f p f f MAX MAX ( ) ( ),0 0, b b b b MAX MAX B (28) Integration of diffusion terms ( ) ( ) ( ) ( ) ( ) ( ) ddd b B b f f s S s n N n W e E e f b n s e Γ Γ Γ Γ Γ Γ Γ Γ Γ δ δ δ δ δ δ (29) In arriving at the above epression, 2 nd order accurate central difference approimation is used for evaluating gradient of scalar at interface. Source term integration S ddd S p f b n s e (30) utting Eqs. (27-30) in Eq. (26), and rearranging, e get the folloing discretied equation.

33 20 ( ) ( ) ( ) ( ) MAX a MAX a MAX a MAX a here b a a a a a a a s s s S n n n N W e e e E B B S S N N W W E E Γ Γ Γ Γ δ δ δ δ,0,0,0,0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t b and t MAX MAX MAX MAX MAX MAX a MAX a MAX a o o b b b f f f s s s n n n e e e b b b B f f f Γ Γ Γ Γ Γ Γ Γ Γ δ δ δ δ δ δ δ δ,0,0,0,0,0,0,0,0 (31) Iterative procedure The process of discretiation converts a partial differential equation into algebraic equation. Hoever, consider solution of u - momentum equation. The scalar represents u velocit component. Since u momentum equation is itself non-linear, the discretied equation remains non-linear. The coefficients a E, etc., contain u velocit making the discretied equation itself non-linear. Hence it cannot be directl solved on a computer. To be able to solve the equation, e use iterative procedure. In this procedure, e assume some value of velocit field, then compute coefficients of Eq. 31 and solve the equation. We get a ne value of velocit. Again compute coefficients of Eq. 31 using latest velocit and solve it. Keep on repeating this procedure till convergence is achieved.

34 7.2.5 Time step or solving Eq. (31), time step has to be estimated. Since the implicit formulation is emploed, ith respect to time step, the code is unconditionall stable. But ith increase in time step, accurac of capturing transient effects reduces. Hence a small time step is desirable from the point of vie of accurac (though not stabilit). The best practice is to follo CL condition for deciding time step. In this condition, the maimum Courant number, defined as vt C should be limited to unit. This practice ill be implemented in solver. Hoever, this constraint ill not be mandator. A choice ill be available to user to perform simulations ith his/her choice of time step Grid fineness The grid generated b user ill decide the values of, and performs mesh independence test to make sure that the grid is sufficientl fine. 8.0 LINEAR EQUATION SOLVER. It is epected that user The above equation is applied to each cell of a computational domain. Thus e get a set of linear algebraic equations. This can be solved b direct (Gauss elimination) or iterative methods (Gauss-Siedel) (Balagurusami 1 ). Direct methods are epensive as large matrices are formed and ana the overall procedure is iterative so there is no point inverting a matri hose coefficients are onl tentative (atankar 17 ). Inverting a large matri again and again involves etremel large computational poer in 3D simulations. Hence iterative solution procedures like Gauss-Siedel are better suited. In CD solver, Gauss-Siedel ith successive over-relaation ill be available. Tri-diagonal matri algorithm, applied to 2D or 3D simulations, gives much better speed of convergence (atankar 17 ). It is combination of Gauss elimination and Gauss-Siedel method. In CD practice, use of TDMA to multi-dimensional simulations is popular. Hence TDMA ill also be available in the solver. Brief description of Gauss-Siedel solver is given here. 8.1 Gauss Siedel Solver The discretied governing equation is ritten in the folloing standard form. a aee aww ann ass a abb b Where subscripts E, W, N, S, and B represent east, est, north, south, front and back neighbours respectivel and stands for centre point and b is its source term. (32) 21

35 The above form is ideall suited for GS solver. The equation is rearranged in folloing form. ( a a a a a a b) E E W W N N S S B B (33) a Then the value of scalar at point is computed assuming some guess value of scalars at all the neighbours. This procedure is repeated till scalars at all points in computational domain are covered. This completes one iteration. At the end of this iteration, one ould get a tentative scalar field hich ould be better than guessed value but not final one. Then the hole procedure of computing scalars at all points of the computational domain is repeated. In each iteration, the latest computed value of neighbor cell is taken for computing center point value. After sufficientl large iterations are performed, the convergence ould be declared. If under-relaation is to be emploed, then folloing form is used (hich is modified form of Eq. 33). aee a W a a a a b W N N S S a α B B ( 1α ) α * a Where α represents under-relaation factor and super-script * represents a previous iteration value. 9.0 STAGGERED ARRANGEMENT O VARIABLES To avoid de-coupling of velocit and pressure, the locations at hich a velocit component is computed is kept half cell distance aa from the location at hich pressure is computed (atankar 17 ). The arrangement in XY plane is shon in ig. 7. The velocit is also kept half cell distance ( / 2 ) aa from pressure location in a similar manner. (34) 22

36 ig. 7 : Staggered arrangement of u and v velocit components ith respect to pressure (in XY plane) 10.0 EQUATION OR RESSURE It should be noted that, for solution of momentum equations, the pressure gradient ould be required. or incompressible flos, there is no direct equation for pressure available. or compressible flo of gases, for eample, equation of state becomes an equation for pressure. But here that facilit is not available. Hence the use of continuit equation is done for this purpose (atankar 17 ). The continuit equation is integrated over a control volume (of scalars). Then the velocities are decomposed into predicted component and a corresponding correction. Then the velocit corrections are related to pressure corrections. Then eventuall an equation for pressure correction is obtained. The hole procedure is ell documented elsehere and hence e directl present the pressure correction equation. The pressure correction field is computed and then added to guessed pressure field to obtain corrected pressure field. The procedure is repeated several times in iterative manner as eplained in 23

37 24 net section. as eplained in net section. ( ) ( ) ( ) ( ) s v s S n v n N u W e u e E B B S S N N W W E E a a a a a a a a here b a a a a a a a, 2, 2, 2, 2 ' ' ' ' ' ' ' ( ) ( ) ab a as an aw ae a a a a a b b B f f, 2, 2 ( ) ( ) ( ) ( ) t v v u u b o b b f f s s n n e e * * * * * * (35) 11.0 ALGORITHM O CD SOLVER 11.1 Semi-Implicit Method for ressure Linked Equations CD methodolog involves the simultaneous solution of a set of coupled non-linear partial differential equations (namel -direction momentum equation, -direction momentum equation, -direction momentum equation, pressure correction equation, equation for turbulent kinetic energ, equation for rate of dissipation of turbulent kinetic energ and thermal energ equation. The partial differential equations are converted into algebraic equations in the process of discretiation. The equations are segregated. Thus one solves for one equation at a time. The nominall linear form is solved b using previous iteration value of the dependent variable. A particular sequence is folloed for solving the set of equations, called algorithm. The solver implements SIMLE (Semi-Implicit Method for ressure Linked Equations) algorithm (atankar 17 ). The algorithm is eplained belo. Step 1 : Assume pressure and velocit fields. Step 2 : Compute predicted value of velocit field (based on assumed pressure field) b solving momentum equations in three directions.

38 Step 3 : Using predicted velocit field, solve pressure correction equation to get field of pressure corrections. Step 4 : Correct pressure and velocit fields. Step 5 : Solve other equations like energ equations, turbulence model equations, etc., if applicable. Step 6 : Check for convergence. If not converged, go back to Step 2 ith corrected velocities and pressures and repeat the hole procedure Convergence Monitoring At step 6, for checking convergence, the value of b in Eq. (35) is monitored. The b term is nothing but mass balance over a given cell. The mass balance in all cells is found out. Then maimum mass balance is found out. If it is greater than the convergence criterion (hich ma be of the order of 1E-7) then iterations are continued, otherise the convergence is declared. Apart from mass balance, the equation residual of each equation is computed from folloing epression. ( a a a a a a b) R a E E W W N N S S B B (36) Convergence is detected if residual of each equation is less than convergence criterion. The other method of convergence monitoring is to monitor spot value. If the spot value reaches a constant value, then convergence is said to be achieved. The user ill get on-screen the values of mass balance and residual of equations being solved. Thus, once the convergence is detected, he/she can inspect the results COMLEX GEOMETRY The solver solves for Cartesian form of equations over a Cartesian grids. The solver is supposed to solve industrial problems hich often involve comple geometries. Consider the solver is applied to solve unstead flo behind a clinder. It is not possible to fit Cartesian mesh to curved boundaries of the clinder. But, ith proper case, this situation can be handled. This matter is eplained here. Since the grid emploed is a 3D Cartesian one, insertion of comple shaped (solid or fluid) regions requires some approimations. This is eplained ith reference to ig. 8 (in 2D for simplicit). 25

39 ig. 8 : Representation of a circle on a Cartesian grid ig. 8 (A) shos the circle ith 88 cartesian grid. A utilit ill mark all rectangular cells hich are completel inside the circle. ig. 8 (B) shos the marked cells for 88 grid. The representation seems to be rather crude. If the non-uniform grid ith larger number of cells (i.e grid shon in ig. 8(C)) is used, then much better representation ould be obtained. After this procedure of identifing the cells in solid region is performed, the numerical technique hich is used to discretie the momentum equations ill adjust the source term of those cells to make sure that in solid regions, the velocit indeed reaches a ero value (atankar 17 ). Consider a scalar transport equation. t u Γ v Γ S (37) This is discretied according to finite volume method. Discretied equation is given belo. ( a C) aee aww a N N ass b (38) Cells in fluid region are given C 0. or the cells hich are in solid region, C is given ver high value, something like -1E20. This makes a ver large and the equation takes the folloing form. ( a a a a b ) E E W W a N N Since a is of the order of 1E20 and numerator is much smaller value than denominator, hence, this leads to 0. S S 26

40 13.0 OST ROCESSING The softare ill be provided ith folloing post-processing capabilities Unstead Results Time variation of an solved variable (u, v,, T, i, k, є) at a probe hose location is defined b user. Time variation of volume eighted temperature of solid regions. Time variation of maimum velocit/temperature/pressure in domain. Animations shoing time evolution of flo fields in an specified plane Stead Results Mass flo rate and average temperature/internal energ/enthalp of flo at each inlet/outlet. This is useful to check integral mass and energ balance. Velocit vector plots, streamlines, contours of pressure, temperature, tke, є, mass fraction of an specie at an plane (hich is aligned along one of the cartesian ais) in the domain. Mass/area/enthalp eighted average of an solved variable on the plane. rofile of an solved for variable along an line specified b user in the domain. The entire data containing all primitive variables ill be available to the user in a file so that he/she can perform post-processing of his/her choice Test Cases The approach of handling comple geometr on a cartesian grid is validated b solving to test cases. A 2D version of the code is developed for this purpose. These test cases are described in forthcoming sections st Test Case : Inclined Lid Driven Cavit Inclined lid driven cavit is an ecellent benchmark test case. The case is solved b various researchers and reliable benchmark data is available for this case (Demirdic 9, Chénier 6, Darish 7, Mathur and Murth 15 ). It is one of the four standard test cases involving nonorthogonalit, proposed b Demirdic et al. 9. The flo in this geometr is solved b CARBS code hich is based on orthogonal cartesian grids. ig. 9 eplains this case. It consists of a cavit ith moving lid at top and three stationar alls out of hich right and left alls are inclined from horiontal at angle θ. Width of the cavit (W) and slanted length (H) are same in all cases considered in this report. VCL represents inclined centre line parallel to side edges. Similarl HCL is the horiontal line at mid-elevation. The u and v velocit profiles 27

41 along these lines, computed using CARBS, are compared ith benchmark data for validation of CARBS Mesh ig. 10 shos a uniform Cartesian 2D grid for the above geometr. After mesh generation, cells hich are completel inside the cavit are marked as fluid cells (red region in ig. 10) and rest are marked as solid cells (blue regions in ig. 10). The fluid and solid cells are separated b staircase boundaries. ig. 9 : Schematic diagram of inclined lid driven cavit 28

42 ig. 10 : Geometr and mesh 15.2 Validation The benchmark results for this case are given b Demirdic et al. (1992), ho performed multi-grid calculations using bod fitted grids ith 2 nd order schemes for convection and diffusion terms of governing equations. The benchmark results for cavit inclined at 30 o and 45 o at Renolds number of 100 and 1000 are given in tabulated form. The top lid velocit is specified to be 1 m/s and cavit idth as ell as inclined height ere taken to be 1 m. Densit is considered to be 1 and viscosit is adjusted to obtain desirable Renolds number Comparison of CARBS results ith Benchmark data at Re 100 and θ45 o Results of the inclined cavit case for Renolds number of 100 at inclination of 45 o are generated and compared ith benchmark data. ig. 11 shos the computed flo field. ig. 12and ig. 13 sho comparison of the computed u and v velocit profiles along vertical and horiontal center lines, respectivel, ith benchmark data. It can be seen that, there is ecellent match beteen computed and benchmark data. 29

43 ig. 11 : lo pattern computed using CARBS (Re100, θ 45 o ) 1.0 Benchmark CARBS Y* Re 100 θ 45 ο u, m/s ig. 12 : U velocit profile along vertical center line for Re100, θ45 o 30

44 0.10 Benchmark CARBS v, m/s Re 100 θ 45 ο ig. 13 : V velocit profile along horiontal center line for Re100, θ45 o Comparison of CARBS results ith Benchmark data at Re 1000 and θ30 o In this case, the cavit skeness as ell as Renolds number is increased. Thus, much more challenging conditions are imposed. CARBS as found to converge ell in this condition also. ig. 14 shos the flo field in cavit. ig. 15 and ig. 16 sho the computed u and v velocit profiles along vertical and horiontal center lines respectivel. The benchmark data is also plotted for comparison and ecellent match is obtained in this case also. * ig. 14 : Re 1000, inclination 30 o 31

45 * Benchmark CARBS Re 1000 θ 30 ο u, m/s ig. 15 : U velocit profile along vertical center line for Re1000, θ 30 o Benchmark CARBS v, m/s Re 1000 θ 30 ο * ig. 16 : V velocit profile along horiontal center line for Re1000, θ30 o 15.3 Convergence characteristics of CARBS To demonstrate convergence behavior of CARBS, the inclination as further increased to 15 o and Renolds number increased to Result is generated on a grid of using ADI 32

46 solver. In this case, convergence as achieved in around 4000 iterations as shon in ig. 17. It can be seen that, convergence on highl skeed cavit at such a high Renolds number is also ecellent. Computational time taken is about 8 minutes on single processor 4 CU Effect of Inclinations on lo attern in Cavit It is interesting to stud the effect of inclination on flo field. or Renolds number of 2000 and for inclinations of 90 o, 75 o, 60 o, 45 o, 30 o and 15 o, flo field is computed using CARBS code. All these results are generated on a grid of using ADI solver. ig. 18 shos the flo field in straight cavit (i.e. inclination from horiontal 90 o ). A large central clockise vorte is visible. To counter rotating vortices are also visible in to loer corners. ig. 19 shos the flo field in cavit ith inclination of 75 o. It can be seen that, the vorte in right loer corner has shrunk. The vorte in left loer corner is bigger. A fourth, ver small vorte is also formed in left loer corner. ig. 20 shos the flo field in cavit ith inclination of 60 o. It can be seen that, the flo field is changed significantl. The vorte attached ith upper moving lid significantl reduces in sie and becomes highl skeed. A large counter rotating vorte sits belo it. The vorte in right loer corner vanishes completel. A third vorte in left loer corner sits belo second vorte. ig. 21 shos the flo field in cavit ith inclination of 45 o. our counter rotating skeed vortices are formed in the cavit. ig. 22 and ig. 23 shos flo field in 30 o and 15 o cavities. It can be seen that, four skeed counter rotating vortices are formed in these cases. But the fluid in left loer corner is almost stagnant. Thus a stagnation one is formed in left loer corner. With increase in inclination, the sie of stagnation one increases as seen from ig. 22 and ig

47 Continuit equation u momentum equation v momentum equation 1E-3 1E-4 Residue 1E-5 1E-6 1E-7 1E-8 1E-9 1E Iterations ig. 17 : Convergence (Re2000, inclination 15 o ) ig. 18 : Streamlines (Re2000, inclination - 90 o ) 34

48 ig. 19 : Streamlines (Re2000, inclination - 75 o ) ig. 20 : Streamlines (Re2000, inclination - 60 o ) 35

49 ig. 21 : Streamlines (Re2000, inclination - 45 o ) ig. 22 : Streamlines (Re2000, inclination - 30 o ) ig. 23 : Streamlines (Re2000, inclination - 15 o ) 15.5 Summar and Conclusions for 1 st Test Case The CARBS code as applied to inclined lid driven cavit problem. or this case, benchmark data is available from literature. The test case offers highl non-orthogonal geometr. Due to formation of multiple vortices, the streamlines are all the places obliquel cut the grid lines. The velocit varies from stagnation to a large value. Hence this test case is ver challenging. 36

50 CARBS as found to converge ell in all cases including highl skeed angle of 15 o. or the 45 o and 30 o inclined cavit at Renolds number of 100 and 1000, computed velocit profiles matches ver ell ith benchmark result nd Test Case lo over Circular/Square Clinder The 2 nd test case chosen as that of eternal flo over a clinder. The flo field around the clinder depends on the inlet Renolds number and is the subject of investigation of present ork. The clinder is subjected to inlet flo. Since eternal flo over clinder is being solved, for sake of computations, a domain is defined b draing three imaginar boundaries as shon in ig. 24. Inlet boundar is at some proper distance upstream the clinder. The top and bottom boundaries are free slip boundaries. A pressure outlet boundar is placed at proper distance donstream the clinder. In conformit ith Sharma 18, the distance L1, L2 and B are computed from folloing epressions. L D, L 16. 5D and B 20D 2, here D is the clinder diameter. Sharma 18 performed numerical simulations of flo around a square clinder. The found that there are three flo regimes depending on inlet Renolds number. At ver lo Renolds number (around 1), attached and stead flo is obtained donstream the clinder. At Renolds number of 2 and above, flo gets separated from clinder but stead flo pattern is obtained. With increase in Renolds number upto 40, the flo is stead but the recirculating vortices behind the clinder keep elongating. Beond Renolds number of 40, vorte shedding behind the clinder starts taking place and flo becomes unstead. Based on vorte shedding frequenc, a non-dimensional number called Strouhl number (St), hich is n D equal to can be defined. Sharma 18 computed Strouhl number for different Renolds u in numbers for square clinder and compared ith benchmark data. Schlichting 19 has given revie of eperimental observations of flo field behind circular clinder. The unstead nature of flo ith vorte formation and shedding is eplained. The effect of Renolds number (based on inlet velocit and clinder diameter) on nd Strouhl number, S, as eplained. It is mentioned that, in Renolds number from V ero to 40, there is no vorte shedding. With increases in Renolds number upto 1000, Strouhl number increases and reaches a value of With further increase in inlet Renolds number upto 6000, Strouhl number remains constant at or 37

51 6000 < Re < 2E5, Strouhl number remains ithin 0.18 to 0.2 approimatel. The data is useful for validation of present test case. ig. 24 : Schematic diagram of flo over a clinder (NOT TO SCALE) 16.1 resent Work A 2D CD code, based on cartesian grid is developed. lo over circular and square clinders over a range of Renolds from 1 to 3000 is computed. The results are compared ith benchmark data Simulation Inputs Various parameters are given in folloing list. Inlet Renolds number 1 to 3000 Working fluid Water (viscosit a-s and densit 1000 kg/m 3 ) Clinder diameter 0.4 m Solver Alternating direction TDMA Grid for clinder 4040 Time step Corresponding to Courant number of unit Under-relaation 0.1 for momentum 16.3 Mesh ig. 25 shos the cartesian mesh mapping the clinder. The grid is clustered around the clinder. It epands in upstream and donstream directions and is finest around clinder. A variable named fluid identifier inde is defined to differentiate fluid and solid cells. The fluid identifier inde is unit for fluid cells and ero for solid cells. ig. 26 shos mesh near 38

52 clinder. The counters of fluid identifier inde are also dran hich indicate the marking of solid and fluid cells. A staircase mesh representing smooth circular boundar is visible. ig. 25 : Non-uniform mesh generation for the domain ig. 26 : Enlarged vie of mesh around clinder 39

53 16.4 Mathematical ormulation The 2D forms of governing equations are derived from Eq. (1) to Eq. (3). Boundar conditions are (1) velocit inlet, (2) pressure outlet, (3) loer and upper free slip boundaries, as shon in ig Code Development A code is developed in ORTRAN 90. The code consists of 1400 lines of listing. The code is developed in modular form ith individual subroutines for computing coefficients of u momentum equation, v momentum equation, pressure correction equation, GS and ADI subroutines, etc. Dnamic memor allocation is implemented for the arras for optimum memor utiliation Mesh Independence The mesh independence test is carried out for square clinder case at Renolds number of 40. The mesh mapping the clinder as increased from 1010 to 2525 to eventuall The remaining mesh in the domain is automaticall generated depending on (i) d and d set for clinder and (ii) domain etents in corresponding directions. The profiles of u, v and p along horiontal line at mid-elevation of domain, as computed b different meshes, are shon in ig. 27, ig. 28 and ig. 29. It can be seen that even coarsest mesh shos negligible deviation from finest mesh considered Square clinder at Re u, m/s mesh 5060 mesh 2525 mesh , m ig. 27 : Variation of u component of velocit along horiontal central line of the domain as computed on different grids 40

54 Square clinder at Re40 mesh 5060 mesh 2525 mesh v, m/s ig. 28 : Variation of v component of velocit along horiontal central line of the domain as computed on different grids, m Square clinder at Re40 mesh mesh 2525 mesh 1010 Gauge static pressure, ascal , m ig. 29 : Variation of static (gauge) pressure along horiontal central line of the domain as computed on different grids 16.7 Time Step Independence or Re200 and for circular clinder, the results are computed ith different time steps. Time step is varied b changing the Courant number. ig. 30 and ig. 31 sho the effect of time 41

55 step on u and p at centre of outlet. It can be seen that, there is some small change in the frequenc of oscillations in u and p. Hence in all the calculations, the Courant number is kept unit Lo Renolds Number Results It is ell knon fact that, at lo Renolds number (upto 40), the flo is stead. Hence the stead state simulations can be done to predict flo field at such a lo Renolds number. The CARBS code, hich is originall an unstead code, can be used in stead state mode in folloing manner. 1) Specif ver large time step e.g. 20 t 10 s. 2) Number of iterations in each time step should be set to a high value. Default value for unstead calculations is 50. This value is set to ) erform just 1 time step calculations C.N. 2 C.N. 1 C.N..5 Circular clinder Re u, m/s time, s ig. 30 : u velocit at center of outlet; as computed b different time steps 42

56 C.N. 2 C.N. 1 C.N..5 Circular clinder Re 200 static gauge pressure, ascal time, s ig. 31 : Static gauge pressure variation at center of outlet; as computed b different time steps The code as successfull able to sho converged stead state results ith above mentioned strateg. or circular clinder at Re40, the convergence histor is shon in ig. 32. All equations ere converged to less than 1E-10 ithin iterations E-3 1E-4 Mass residue U momentum eqn residue V momentum eqn residue ressure correction eqn residue 1E-5 Residue 1E-6 1E-7 1E-8 1E-9 1E-10 1E-11 1E Iterations ig. 32 : Convergence histor (Circular clinder at Re40) 43

57 or square and circular clinders, the results are generated for Renolds number of 1, 10, 20, 40 and Circular clinder results ig. 33 shos flo field generated for circular clinder at Re1. It can be seen that, the flo is attached ith the boundar of clinder. ig. 34 shos flo field generated at Re10. lo separation is visible. To re-circulating vortices are visible. The flo field is smmetric and stead. ig. 35 shos flo field generated at Re20. Qualitativel, its same as that at Re 10. The length of re-circulating region is increased. With further increase in Renolds number to 40, as shon in ig. 36, length of the to re-circulating vortices increases further. ig. 37 shos the vorticit contours. Vorticit is generated at the stagnation point and convected along the flo. ig. 38 shos the computed pressure distribution and velocit vectors in the sstem. With increase in inlet Renolds number, the point of separation changes. ig. 33 : lo field around circular clinder at Re 1 44

58 ig. 34 : lo field around circular clinder at Re 10 ig. 35 : lo field around circular clinder at Re 20 45

59 ig. 36 : lo field around circular clinder at Re 40 ig. 37 : Vorticit field around circular clinder at Re 40 46

60 ig. 38 : ressure and velocit fields (Circular clinder Re 40) Square clinder results ig. 39 shos computed flo field around square clinder at Re 1. ig. 40 shos the enlarged vie of flo field near the clinder. The flo is almost attached ith the square clinder. At Re10, as shon in ig. 41, flo is separated. To smmetric re-circulating vortices are visible. With increase in Renolds number, as shon in ig. 42 and ig. 43, the re-circulating vortices elongate in the direction of flo. Due to sharp edges in this geometr, the point of separation is same i.e. corner of square. ig. 44 shos the pressure distribution and velocit vector for Re40 case. 47

61 ig. 39 : lo field around square clinder at Re 1 ig. 40 : lo field around square clinder at Re 1 48

62 ig. 41 : lo field around square clinder at Re 10 ig. 42 : lo field around square clinder at Re 20 49

63 ig. 43 : lo field around square clinder at Re 40 ig. 44 : ressure and velocit field around square clinder at Re 40 50

64 16.9 Vorte Shedding at High Renolds Number The code as run in stead state mode (as described earlier) for square clinder case. The inlet Renolds number is set to It as found that, after iterations, the code gives un-smmetric result. The computed flo field is shon in ig. 45. The convergence is shon in ig. 46. It can be seen that the code is diverging. This indicates that, at Re62.5, the flo no more remains stead. The flo field at higher Renolds number is then computed in purel transient manner. ig. 45 : lo field around square clinder at Re

65 Mass X-Momentum 1E-3 1E-4 Equation residue 1E-5 1E-6 1E-7 1E-8 1E-9 1E-10 1E Iterations ig. 46 : Convergence histor for Re 62.5 (stead state mode) ig. 47 shos the computed flo field at different instances of time for square clinder at Re100. It can be seen that the flo field is cclic. A small vorte is visible in 1 at top right corner. Then vorte gros as shon in 2 and 3. At 4, it gets detached from the clinder. At 5, a similar vorte is formed at bottom right corner. At 6 and 7, the vorte gros and eventuall at 8, it gets detached from clinder b hich time, net vorte gets generated again at right top corner thus repeating entire ccle. The vortices generated at the to sharp corners keeps dissipating in the donstream. Thus vorte shedding is computed. The pressure field also keeps cclicall changing ith time. The lift force, defined as pressure difference on top and bottom surface of clinder multiplied b area, keeps cclicall changing as shon in ig. 48. requenc of vorte shedding is computed from the frequenc of lift force variation. ig. 49 shos the variation of Strouhl number ith inlet Renolds number. The results are compared ith eperimental data. There seems to be satisfactor agreement of computed result ith eperimental data. A similar plot for circular clinder is shon in ig. 50. It can be seen that, over a ide range of Renolds number, the computed result is match ver ell ith benchmark data. It ma be noted that mesh of 2020 does not give eact result. or proper result, one needs 4040 grid to map clinder. 52

66 ig. 47 : streamlines at different times (square clinder at Re 100) 53

67 Square clinder at Re Lift force, N lo time, s ig. 48 : variation of lift force ith time CARBS Ept. (Robichau et al.) Square Clinder Strouhl number Renolds number ig. 49 : Variation of Strouhl number ith inlet Renolds number for square clinder 54

68 Strouhl number Benchmark data (Schlichting) CARBS - Circular clinder Renolds number ig. 50 : Variation of Strouhl number ith inlet Renolds number for circular clinder 17.0 Comparison of erformance of GS and ADI Solvers Comparison of GS and ADI (Alternating direction TDMA) is performed. Simulation as run for 250 time steps. The flo over circular clinder at Renolds number of 100 as computed. All other simulation parameters (grid, under-relaation, number of seep, etc) ere kept the same in both the cases. Convergence histor in initial 20 time steps is shon in ig. 51. The convergence histor for 200 th to 220 th time step is shon in ig. 52. It can be seen that, overall alternating direction TDMA orks better than Gauss Siedel. In later part, GS performs good job. B increasing the number of iterations, its convergence can be brought to the level of ADI. But initiall, the residue as found to saturate in each time step. Convergence of ADI as 2 orders of magnitude better. In computing time accurate flos, the ADI ill give much better results than GS. The time taken b ADI as bit higher than GS. 55

69 GS ADI 1E-3 Mass residue 1E-4 1E-5 1E-6 1E Iteration ig. 51 : Convergence b GS and ADI schemes in initial 1000 iterations 1E-4 GS ADI 1E-5 Mass residue 1E-6 1E-7 1E Iteration ig. 52 : Convergence b GS and ADI schemes in later time period 56

70 18.0 Summar and Conclusions ormulation for 3D transient CD softare is developed. Various issues like handling comple geometr, discretiation, turbulence modeling, heat transfer modeling, thermophsical properties, etc, ere discussed. A 2D test code as developed and applied to to test problems. 1 st test case is that of lid driven flo in inclined cavit. 2 nd test case is the flo over clinder. The 1 st test case involved stead internal flo subjected to WALL boundaries. The 2 nd test case involved unstead eternal flo subjected to INLET, OUTLET and REE- SLI boundar tpes. In both the test cases, non-orthogonal geometr as involved. It as found that, under such a ide conditions, the cartesian grid based code as found to give results hich ere matching ith benchmark data. Based on this, development of 3D general purpose code based on the proposed approach can be taken up. 57

71 Bibliograph 1) Balagurusam E., Numerical Methods, Tata McGra-Hill Education rivate Limited, Ne Delhi, ) Balagurusami E., rogramming in ANSI C, 6 th Edition, Tata McGra-Hill Education rivate Limited, Ne Delhi, ) Bird R. Bron, Steart W. E. and Lightfoot Edin N., Transport henomenon, John Wile & Sons (SEA) te. Ltd, Singapore, ) Bisas G. and Esaran V., Turbulent los : undamentals and Modelling, Narosa ublishing House, Ne Delhi, ) Cengel Y. A. and Ghajar A. J., Heat and Mass Transfer, 4 th Edition, Special Indian Edition, Tata McGra-Hill Education rivate Limited, Ne Delhi, ) Chénier E., Emard R. and Touai O., Numerical Results Using a Colocated inite- Volume Scheme on Unstructured Grids for Incompressible luid los, Numerical Heat Transfer, art B: undamentals 49 (2006) : ) Darish M., Sraj I. and Moukalled., A Coupled Incompressible lo Solver on Structured Grids, Numerical Heat Transfer, art B: undamentals, 52 (2007) :: ) Date A. W., Introduction to Computational luid Dnamics, Cambridge Universit ress, ) Demirdic I., Lilek Z., and eric M., luid lo and Heat Transfer Test roblems for Non-Orthogonal Grids: Bench-Mark Solutions, Int. J. Numer. Meth. luids, 15 (1992) :: ) eriger J. H. and eric M., Computational Method for luid Dnamics, Springer ublication, ) o R. W. and McDonald A. T., Introduction to luid Mechanics, 4 th edition, John Wile and Sons, ) Incropera.. and DeWitt D.., undamentals of Heat and Mass Transfer, 5 th Edition, John Wile and Sons (Asia) te Ltd, ) Kanetkar Y., Let Us C, 6 th Edition, BB ublications, Ne Delhi, ) Launder B. E. and Spalding D. B., 1974, The Numerical Computation of Turbulent los, Computational Methods in Applied Mechanics and Engineering, 3 (1974) ::

72 15) Mathur S. R. and Murth J. Y., A ressure-based Method for Unstructured Meshes, Numerical Heat Transfer, art B: undamentals,. 31 (1997), :: ) Murlidhar K. and Sundararajan T., Computational luid lo and Heat Transfer, 2 nd Edition, Narosa ublishing House, ) atankar Suhas V., Numerical Heat Transfer and luid lo, Hemisphere ublishing Corporation, ) Sharma A. in Murlidhar K. and Sundararajan T., Computational luid lo and Heat Transfer, 2 nd Edition, Narosa ublishing House, ) Schlichting H., Boundar Laer Theor, McGra Hill book compan, 7 th edition, ) Som S. K. and Bisas G., Introduction to luid Mechanics and luid Machines, 2 nd Edition, Tata McGra-Hill ublishing Compan Limited, Ne Delhi, ) Rajaraman, arallel Computers : Architectures and rogramming, HI Learning vt. Ltd., Ne Delhi,

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