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SIEMENSSIEMENSSIEMENS

Simcenter FlowSolver ReferenceManual

Simcenter • 11

Contents

Proprietary & Restricted Rights Notice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

Mass and momentum equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2Low speed equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3High speed equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

Scalar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3Scalar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3Modeling a passive scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4Gas mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4Humidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6

Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8Ideal gas law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10Redlich-Kwong real gas equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11Non-Newtonian fluid models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11

Turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

Turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1Fixed viscosity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2Mixing length turbulence model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2k-ε model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3k-ω model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4SST model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8

Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8Prandtl mixing length hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8Local equilibrium assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10Launder and Spalding's eddy viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10Further Considerations for the k-ε Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11

Source terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

Buoyancy force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1Buoyancy force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1Enclosed cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3Cavities with openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4

Centripetal and Coriolis forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4Flow resistance through porous materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5

Simcenter 11 Simcenter Flow Solver Reference Manual 3

Contents

Flow resistance through porous materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5Heat sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7Condensation and evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7

Condensation and evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7Mass and scalar conservation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7Energy conservation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1Solid walls and flow surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1Slip wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1No-slip wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1Extensions of the momentum wall function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6Thermal wall function for high speed flows and flows with viscous heating . . . . . . . . . . . . . 5-8General heat transfer coefficient for natural convection . . . . . . . . . . . . . . . . . . . . . . . . . 5-11

Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19Fan curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19Flow angle on fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-26Swirl on fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-27Energy equation in fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28Turbulence quantities on fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28Humidity and scalar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29

Openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-30Convective outflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-33Supersonic inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34

Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34Thin perforated plate screen correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-35Wire screens correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-37Silk thread screens correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-38

Symmetry boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-38Periodic boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-38

Particle tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

Lagrangian particle tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2Modeling of the drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4Modeling of the perturbed flow field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6Modeling of chaotic motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10Boundary treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12Wall boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13Inlet and symmetry boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15Periodicity boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16Initial conditions for injected particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17

Cunningham correction factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18Sutherland's law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-20

4 Simcenter Flow Solver Reference Manual Simcenter 11

Contents

Contents

Particle tracking limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21

Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

Control-volume method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1Details of the discretization for the mass conservation equation . . . . . . . . . . . . . . . . . . . . . . . 7-3Details of the discretization for the other conservation equations . . . . . . . . . . . . . . . . . . . . . . 7-3Convective terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

Convective terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4First order scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6Higher order bound schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6Higher order schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6Flux limiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

Diffusion terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8Source terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9Pressure term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9Rotating frames of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9Mixing Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-11

Solver numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

Solver numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

Temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

Temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index-1


Contents

Proprietary & Restricted Rights Notice

© 2016 Siemens Product Lifecycle Management Software Inc. All Rights Reserved. This softwareand related documentation are proprietary to Siemens Product Lifecycle Management Software Inc.

The Simcenter Flow solvers including the NIECE parallel flow solver and related documentation areproprietary to Maya Heat Transfer Technologies LTD.

All other trademarks are the property of their respective owners.

MayaMonitor is based in part on the work of the Qwt project (http://qwt.sf.net).

Trilinos Copyright and License

The parallel flow solver NIECE uses the Trilinos library. Trilinos, Copyright (2001) SandiaCorporation, is a free software licensed under the terms of version 2.1 of the GNU Lesser GeneralPublic License as published by the Free Software Foundation. A copy of the license is found inthe[software_installation]\NXCAE_extras\tmg\install\GNU_LGPLicense.txt file of your distributionand you can read it at http://www.gnu.org/licenses/lgpl.html. The Trilinos library may be obtained fromSandia National Labs at http://trilinos.sandia.gov/. For a period of up to three years after the releasedate of this software, licensed users can contact MAYA Heat Transfer Technologies LTD. to obtain thesource code of Trilinos used with this release, as well as the object code required to link NIECE with acustomized version of Trilinos. To obtain this code, please email [email protected].

METIS Copyright and License

The parallel flow solver NIECE uses the METIS library with permission. METIS, Copyright (1997)The Regents of the University of Minnesota. A copy of the METIS reference manual is located at[software_installation]\NXCAE_extras\tmg\install\metis-manual.pdf in your distribution. Relatedpapers are available at http://www.cs.umn.edu/~metis, while the primary reference is “A Fast andHighly Quality Multilevel Scheme for Partitioning Irregular Graphs”. George Karypis and Vipin Kumar.SIAM Journal on Scientific Computing, Vol. 20, No. 1, pp. 359-392, 1999.

MPICH2

Copyright Notice+ 2002 University of Chicago

Permission is hereby granted to use, reproduce, prepare derivative works, and to redistribute toothers. This software was authored by:

Mathematics and Computer Science DivisionArgonne National Laboratory, Argonne IL 60439

(and)

Department of Computer ScienceUniversity of Illinois at Urbana-Champaign

GOVERNMENT LICENSE


http://qwt.sf.net

http://www.gnu.org/licenses/lgpl.html

http://trilinos.sandia.gov/

mailto:[email protected]

http://www.cs.umn.edu/~metis


Portions of this material resulted from work developed under a U.S. Government Contract and aresubject to the following license: the Government is granted for itself and others acting on its behalf apaid-up, nonexclusive, irrevocable worldwide license in this computer software to reproduce, preparederivative works, and perform publicly and display publicly.

DISCLAIMER

This computer code material was prepared, in part, as an account of work sponsored by an agencyof the United States Government. Neither the United States, nor the University of Chicago, norany of their employees, makes any warranty express or implied, or assumes any legal liability orresponsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privately owned rights. Portions ofthis code were written by Microsoft. Those portions are Copyright (c) 2007 Microsoft Corporation.Microsoft grants permission to use, reproduce, prepare derivative works, and to redistribute toothers. The code is licensed "as is." The User bears the risk of using it. Microsoft gives no expresswarranties, guarantees or conditions. To the extent permitted by law, Microsoft excludes the impliedwarranties of merchantability, fitness for a particular purpose and non-infringement.

8 Simcenter Flow Solver Reference Manual Simcenter 11


Chapter 1: Introduction

In Simcenter Flow, the flow solver computes a solution to the non-linear, partial differential equationsfor the conservation of mass, momentum, energy, and general scalars in general complex 3Dgeometries. It uses an element-based finite volume method and a coupled algebraic Multigrid methodto discretize and solve the governing equations.

Physical models include laminar or turbulent, incompressible or compressible flow, natural convection,rotating frame of reference, non-newtonian fluids, porous blockages, mixtures, humidity, condensationand evaporation at walls, and general boundary conditions for fluid flow and heat transfer. Details ofthe mathematical model, the discretization of the equations, and of the solution method used in theSimcenter Flow solver are presented in the following sections of this document.

Simcenter 11 Simcenter Flow Solver Reference Manual 1-1

Chapter 2: Governing equations

Mass and momentum equationsThe mass and momentum equations, when expressed in Cartesian coordinates and using thetensorial notation [1, 2, 3], are:

Equation 2-1.

Equation 2-2.

In Eqs. (2-1) and (2-2), the Einstein convention is used (i, j, k = 1,2,3), Uj and uj are the componentsof the mean and the fluctuating velocity in the xj direction, P is the pressure, ρ is the density ofthe fluid, μ is the dynamic viscosity of the fluid, and represents the turbulent (or Reynolds)stresses. Sm and are the source terms for the mass and momentum equations, respectively.

Equation (2-1) expresses the conservation of mass of the fluid and is valid for incompressible andcompressible flows.

Equation (2-2) represents the conservation of momentum for general flows. The various terms ofthis equation are in the same order of the equation: the transient term, the convection term, thepressure gradient term, the stress term and the source term. This equation is also valid for both,incompressible, and compressible flows.

The source term in Eq. (2-2) can represent body forces, or flow resistance forces:

1. For natural convection flows, includes the buoyancy force.

See Buoyancy force for more information.

2. For flows in a rotating frame of reference, it includes Coriolis and centripetal forces.

See Centripetal and Coriolis forces for more information.

3. For flows through porous blockages, contains additional resistance terms.

See Flow resistance through porous materials for more information.



Energy equation

Energy equation

In order to describe completely deferent kinds of flows, the equation of conservation of energymust also be implemented. Starting from total energy equation, the high speed form and the lowspeed form of the energy equation will be derived.

The instantaneous total energy equation in tensorial notation is [1]

Equation 2-3.

where e is the internal energy, U, the velocity magnitude, qi, the heat flux in the direction xi, and q', aheat generation or heat sink per unit volume.

In Eq. (2-3), the various terms represent, in order: the rate of energy gain per unit volume, the rateof energy input per unit volume due to convection, the rate of energy addition due to conductionand turbulent mixing, the rate at which work is done on the fluid by pressure and viscous forces(dissipation term), and the rate of heat generation by internal sources.

The second and fourth terms in the above equation can be combined as:

Equation 2-4.

Furthermore, using the definition of the total enthalpy, ho,

Equation 2-5.

and combining with Eq. (2-3) into Eqs. (2-4) and (2-5) gives

Equation 2-6.

where Sh is the energy source term. In Eq. (2-5) h is the static enthalpy of the fluid.

2-2 Simcenter Flow Solver Reference Manual Simcenter 11


Governing equations

Low speed equation

For low speed flows (Mach < 0.3), the energy equation is simplified for robustness and for round-offpurposes. For low speed incompressible and compressible flows, the pressure work and dissipationterms in Eq. (2-6) can be neglected. The simplified form of the energy equation has the mechanicalenergy subtracted from the total energy, and becomes a thermal energy equation [2].

Equation 2-7.

This form of the equation ensures conservation of the thermal energy, and avoids round-off problems.The low speed form of the energy equation is used by default in the flow solver.

High speed equation

After modeling the conduction and taking the Reynolds average of Eq. (2-6) [1], Eq. (2-6) becomes:

Equation 2-8.

where:

• k is the thermal conductivity.

• cp, the specific heat at constant pressure.

• h' is the fluctuating static enthalpy.

• is the turbulent or Reynolds flux.

This equation expresses the conservation of the total energy (i.e. the thermal energy plus themechanical energy). It is valid for all flow situations, but the user should limit it for high speed flows.

Scalar equation

Scalar equation

In addition to the mass, momentum, and energy equations, the flow solver also solves one or moregeneral scalar equations that have the following form:


Governing equations


Equation 2-9.

where the mass fraction is defined as:

Equation 2-10.

ρa, ρΦ, and ρ represent the density of the main fluid, the density of the scalar and the density of

the fluid mixture, respectively. DΦ is the scalar diffusion coefficient and is the turbulent orReynolds flux where Φ' is the fluctuating mass fraction of the scalar.

The left hand side terms of Equation 2-9 are the transient term and the convective term, respectively,and the right hand side term is the diffusion term.

The general scalar equation is used for:

• Modeling a passive scalar

• Modeling a gas mixture

• Modeling humidity

Modeling a passive scalar

The general scalar equation can be used to model a passive scalar. In this case, it is assumed thatthe second component is present in the main fluid in very small quantities:

• Φ << 1

• ρ ≈ ρa

The passive scalar does not have any influence on the flow. The fluid properties are that of the mainfluid, and the properties of the second component are not needed.

Gas mixtures

Modeling a gas mixture

Scalar equation can be used to model a gas mixed in any proportion with the main gas. The twogases are assumed to behave as perfect gases using the ideal gas law (see Ideal gas law for moreinformation).

Equations 2-1, 2-2, and 2-7or 2-8 represent the conservation of mass, momentum, and energy forthe gas mixture. The fluid properties calculated by the flow solver are the properties of the gasmixture. In order to calculate these mixture properties, the user must supply the properties of thesecond gas (scalar).



Governing equations

When modeling a gas mixture, the density, ρ, the specific heat at constant pressure, Cp, the thermalconductivity, k, and the dynamic viscosity, μ of the gas mixture are calculated at every iteration. Thisproperty update is based on the assumption that the two gases behave as perfect gases.

Equation 2-11. Ideal gas equation of state forgas 1

Equation 2-12. Ideal gas equation of state forgas 2

where p1, p2, R1, and R2 represent the partial pressures of gas 1 and of gas 2, and the gas constantsof the gas 1 and gas 2, respectively.

Pressure and density

The pressure and density of the gas mixture are given by:

Equation 2-13. Pressure of the gas mixture Equation 2-14. Density of the gas mixture

Specific heat at constant pressure of the gas mixture

The specific heat of the mixture, Cp, is calculated from the following equations [18]:

Equation 2-15. Specific heat of the gas mixture

Given the molar masses of gas 1, , and gas 2, , the following table defines and .

Property Name Equation

The molar specific heat of themixture

Equation 2-16.

Themolar mass of the mixture

Equation 2-17.

Thermal conductivity and dynamic viscosity of the gas mixture

The thermal conductivity, k, and the dynamic viscosity, μ, of the mixture are calculated using themethod of Wilke [19] which is valid for gases at low pressures. According to this method


Governing equations


Equation 2-18.

Equation 2-19.

Equation 2-20.

where ηm, η1, and η2 represent the property (thermal conductivity or viscosity) of the gas mixture, ofgas 1, and of gas 2, respectively, and y1 and y2 are the mole fractions of the two gases given by:

Equation 2-21. Equation 2-22.

Humidity

Modeling humidity

Humidity (water vapor in air) is modelled using the gas mixture scalar equation.

Rather than specifying the boundary and initial conditions in terms of mass ration, in this case,they are specified either in terms of:

• Relative humidity

• Specific humidity

The preprocessor converts the humidity values to mass ratio values. Once the solution of theconservation equations is obtained, the postprocessor coverts the mass ratio values back to relativeand specific humidity values.

Furthermore, the water vapor properties do not need to be supplied, as they are readily available inthe flow solver.

Conversion from relative humidity to mass ratio

By definition, the relative humidity, φr, is the ratio of the partial pressure of the water vapor, pv, to thesaturation pressure of the water vapor at a given temperature, pv. sat(T):



Governing equations

Equation 2-23.

A value of φr equal to 100% indicates the onset of condensation. It can be seen from Equation 2-23that the air/vapor mixture gets closer to the condensation state if water vapor is added to the fluid (thepartial pressure of water vapor then increases) or if the temperature goes down.

The analytical formulas proposed in [13] are used to calculate the water vapor saturation pressure.They are:

For -100°C < T < 0°C

ln(pv. sat) = C1/T + C2 + C3T + C4T2 + C5T3 + C6T4 + C7ln(T)

Equation 2-24.

with

Coefficient Value Coefficient ValueC1 5.6473590 x 10+3 C2 6.3925247C3 9.6778430 x 10–3 C4 6.2215701 x 10–7C5 2.0747825 x 10–9 C6 - 9.4840240 x 10–13C7 4.1635019

For 0°C ≤ T< 200°C

ln(pv. sat) = C8/T + C9 + C10T + C11T2 + C12T3 + C13ln(T)

Equation 2-25.

with

Coefficient Value Coefficient ValueC8 - 5.8002206 x 10+3 C9 1.3914993C10 - 4.8640239 x 10-2 C11 4.1764768 x 10–5C12 - 1.4452093 x 10–8 C13 6.5459673

where the temperature T in Equations 2-24 and 2-25 is in Kelvin and pv. sat, in Pascal.

For given values of T and φr, the water vapor partial pressure is calculated from Equations 2-24and 2-25. Given the pressure of the water vapor/air mixture, p, the mass ratio is then obtained bycombining Equations 2-10 to 2-14 where the gas 1 properties are air properties and gas 2 propertiesare water vapor properties:

Equation 2-26.

where:


Governing equations


• Rv is the water vapor gas constant.

• Ra is the air's gas constant.

Conversion from specific humidity to mass ratio

By definition, the specific humidity φs is the ratio of the water vapor density to the dry air density, i.e.:

Equation 2-27.

The conversion from specific humidity to mass ratio is simply obtained from

Equation 2-28.

Equations of state

Equation of state

An equation of state is a constitutive equation which provides a mathematical relationship betweenthermodynamic state variables for a given material. With the mass, momentum, and energyequations, it completes the mathematical representation of your fluid model.

The following state variables need to be defined:

• Density, ρ

• Dynamic viscosity, μ

• Specific heat a constant pressure, cp

• Conductivity, k

• Specific enthalpy, h

You define all of these material variables except the specific enthalpy in this software. Thesematerial variables can vary with time, temperature, pressure, or they can vary with both temperatureand pressure (bivariate properties).

In the case of bivariate properties, and in the absence of the standard state models such as ideal gas,the flow solver reads the user-specified bivariate table, and then performs a bi-linear interpolationwith respect to temperature and pressure in order to calculate the value of the state variables.

The flow solver uses the state variables differently depending if your fluid is a gas or a liquid.



Governing equations

Note

The flow solver differentiates between a liquid and a gas as follows:

• If the gas constant, Rs, is defined, the fluid material is a gas.

• If the gas constant, Rs, is not defined, the fluid material is a liquid.

When your fluid is a liquid, the density ρ and the specific heat cp are assumed constant. The flowsolver calculates the specific enthalpy, h, from the energy equation Eq. (2-6), and uses the followingequation to calculate the temperature T.

h=cpT+P/ρ

Equation 2-29.

When your fluid is a gas, an equation of state is used to model the relationship betweenthermodynamic state variables.

The flow solver supports the following models:

• Ideal gas law

• Redlich-Kwong real gas equation of state

The flow solver calculates the specific enthalpy, h, from the energy equation Eq. (2-6), and uses thefollowing equation to calculate the temperature T:

h=cpT

Equation 2-30.

where cp is the specific heat at constant pressure of the gas.


Governing equations


Ideal gas law

The ideal gas law equation of state used by the flow solver is given by:

P = ρRsT

Equation 2-31.

where:

• P is the pressure of the gas.

• T is the temperature of the gas.

• ρ is the density of the gas.

• Rs is the specific gas constant (J/(kg·K) in SI units).



Governing equations

Redlich-Kwong real gas equation of state

The Redlich-Kwong real gas equation of state is given by:

Equation 2-32.

where:

• P is the pressure of the gas.

• T is the temperature of the gas.

• ρ is the density of the gas.

• Vm is the molar volume of the gas.

• R is the universal gas constant (8.314472 J/(mol·K)).

The constants a and b are defined as:

a=0.42748(R2TC2.5/PC)

b=0.08662(RTC/PC)

Equation 2-33.

where:

• PC is the critical pressure of the gas.

• TC is the critical temperature of the gas.

The Redlich-Kwong equation of state is more realistic than the ideal gas law at high pressure andvalid when:

Equation 2-34.

Non-Newtonian fluid models

Non-Newtonian fluids

In non-Newtonian fluids, the shear stress of the fluid is not proportional to the rate of deformation,therefore viscosity is no longer constant. An additional model is required to model viscosity, you canuse one of the following models according to the behavior of the fluid:

• Power-Law model


Governing equations


• Herschel-Bulkley model

• Carreau model

Power-Law model

Equation 2-35 presents the fluid viscosity, μ, of a Power-Law fluid defined in Simcenter Flow.

Equation 2-35.

with

μ min < μ < μmaxEquation 2-36.

where:

• K is the consistency index.

• is the shear rate.

• n is the power law index.

• T0 is the reference temperature.

• T is the fluid temperature.

• μ min is the minimum viscosity limit.

• μmax is the maximum viscosity limit.

Herschel-Bulkley model

Equations 2-37 and 2-38 present how the fluid viscosity, μ, of a Herschel-Bulkley fluid is modelled inSimcenter Flow.

Equation 2-37.

Equation 2-38.

where:

• K is the consistency index.



Governing equations



• τ0 is the yield stress.

• τint is the intersection value for which Equations 2-37 and 2-38 are equal.

• μ0 is the yield viscosity or plastic viscosity.

Figure 2-1. Herschel-Bulkley model for shear rate

To smooth convergence, a fluid define with Herschel-Bulkley model, will behave like a Newtonianfluid until the local shear stress reaches intersect (A), shown as the red curve. The yield viscosity,μ0, is a mathematical artefact introduced to improve convergence of the Herschel-Bulkley modelwhen the stress is less than the yield stress.

The blue curve shows the non-Newtonian Herschel-Bulkley behavior after intersect (A).

Carreau model

Equation 2-39 presents the fluid viscosity, μ, of a Carreau fluid defined in Simcenter Flow.

Equation 2-39.

where:

• λ is the time constant.



• T0 is the reference temperature.

• T is the fluid temperature.


Governing equations


• μ∞ is the infinite-shear viscosity.

• μ0 is the zero-shear viscosity.



Chapter 3: Turbulence models

Turbulence modelsIn Simcenter Flow, the flow field can be solved as laminar or as turbulent. Five turbulence models areavailable to model the Reynolds (or turbulent) stresses and fluxes:

1. Fixed viscosity model

2. Mixing length turbulence model

3. Standard two-equation k-ε model

4. Wilcox k-ω model

5. Shear stress transport model (SST)

Equations (2-1) to (2-9) are valid for laminar or turbulent flows.

For laminar flows, the Reynolds stresses, , and the Reynolds fluxes, and ,are simply zero.

With all five models, the Reynolds stresses and fluxes are evaluated using a Boussinesq eddyviscosity assumption [4] i.e.

Equation 3-1.

Equation 3-2.

Equation 3-3.

Equation 3-4.



where k is the turbulence kinetic energy, Prt is a turbulent Prandtl number, and Sct , the turbulentSchmidt number.

In the code, the right term in Eq. (3-1) ((-2/3)ρkδij) is included in the pressure term of the momentumequation and is not calculated explicitly.

In addition to these turbulence models, Simcenter Flow can also use Large Eddy Simulation (LES)turbulence model by solving the filtered Navier-Stokes equations to resolve large eddies when yousimulate turbulent fluid flow.

Fixed viscosity modelFor the fixed viscosity model, μt, the turbulent viscosity, is evaluated from:

μt=0.01ρVml

Equation 3-5.

where Vm is the mean flow velocity scale and l is a turbulent eddy length scale.

When not provided by the user, Vm is computed as the maximum flow speed in the domain, and l istaken as 1/7 of the cube root of the volume of the domain.

Mixing length turbulence modelThe mixing length turbulence model is a zero-equation model which uses the following relationshipto calculate the turbulent viscosity:

Equation 3-6.

where l is the mixing length and S is the modulus of the mean strain rate.

The mixing length l and damping factor fl are defined as:


κ is the Von Karman constant (κ =0.41), y is the normal distance from the node to the wall and ymaxis a characteristic length scale for the model. If this length scale is not specified by the user, adefault value is computed using

Equation 3-9.



Turbulence models

where Vol is the volume of the fluid domain and Awetted is the wetted area. A length scale is computedfor each fluid domain in the model. Note that for a square or cylindrical duct, the above lengthscale is equivalent to half the hydraulic diameter.

For internal nodes (i.e. nodes which are not touching a wall), the modulus of the mean strain rateis given by:


For wall nodes, with u* as the shear velocity, the strain rate is based on the logarithmic wall function:

Equation 3-12.

k-ε modelWith the standard two-equation k-ε model, the turbulent viscosity is evaluated from

Equation 3-13.where kis the turbulent kinetic energy, ε is the dissipation rate of the turbulent kinetic energy, Cμis a constant, and ρ is the density of the fluid.

The turbulent kinetic energy, k, and the dissipation rate of turbulent kinetic energy, ε, are obtained bysolving a conservation equation for each of these two quantities. Those equations are

Equation 3-14.

Equation 3-15.where the effective diffusion coefficients of k is:

Equation 3-16.


Turbulence models


the effective diffusion coefficients of ε is:

Equation 3-17.and Pk is the production rate of the turbulent kinetic energy defined as:

Equation 3-18.The constants in equations (3-13) to (3-18) are

Prt = 0.9; Cμ = 0.09; Cε1 = 1.44; Cε2 = 1.92; σk = 1.0 and σε = 1.3.

k-ω modelWith the standard k-ω turbulence model, the turbulent viscosity is given as

Equation 3-19.where k is the turbulent kinetic energy, ω is the specific dissipation rate of the turbulent kineticenergy, and ρ is the density of the fluid.

The turbulent kinetic energy, k, and the specific dissipation rate of turbulent kinetic energy, ω, areobtained by solving a conservation equation for each of these two quantities. The equation for k is

Equation 3-20.The equation for ω is

Equation 3-21.where the effective diffusion coefficient of k is:

Equation 3-22.and the effective diffusion coefficient of ω is:



Turbulence models

Equation 3-23.

and Pk is the production rate of turbulent kinetic energy.

The quantities β and β* are defined as β =β0fβ and β* =β*0fβ* where

Equation 3-24.

and

Equation 3-25.

The constants in equations (3-19) to (3-25) are:

β0 = 0.072; β*0 = 0.09; a = 0.52; σk-ω = 2 and σω = 2

Sij is defined by equation (2.33) and Ωij as follows:

Equation 3-26.

SST modelWith the shear stress transport (SST) turbulence model the turbulent viscosity is:

Equation 3-27.

Equation 3-28.

where the equations for k and ω are:


Turbulence models


Equation 3-29.

Equation 3-30.

When F1 = 0, the transport equations are equivalent to the k-ε model, and when F1 = 1, thetransport equations are equivalent to the k-ω.

The blending function is given by:

Equation 3-31.

where

Equation 3-32.

with

Equation 3-33.

and y is the distance from the wall.

The second blending function is given by:

Equation 3-34.

with

Equation 3-35.



Turbulence models

A production limiter is used to prevent build-up of turbulence in stagnation regions:

Equation 3-36.

with

Equation 3-37.

TKELIM is 10 when no wall function and big number (default) when wall function is used.

The modulus of the mean strain rate is given for interior nodes by:

Equation 3-38.

Equation 3-39.

and for wall nodes by:

Equation 3-40.

The following table list the various coefficients that are given as blended constants.




The following table lists the constants for this model.

Coefficient Value Coefficient Value


Turbulence models


0.09 2 0.0828

1 0.075 1 5/9

2 0.44 σk1 2σω1 2 σk2 1σω2 1/0.856 a1 0.3

Assumptions

Assumptions

The derivation of the near-wall relations for turbulent flows depends on several assumptionspresented in this section. We will assume that the flow close to the wall is in the x direction.

Fully developed flow

This means that all streamwise derivatives vanish, i.e. , and that the flow is parallel to the wall.

Constant shear layer flow

This is equivalent to requiring that there is no pressure gradient or other momentum source term.

Equation 3-47.

Boussinesq or eddy viscosity assumption

This states that the Reynolds stress can be expressed as the product of an effective eddy viscosityand the mean flow strain rate.

Combining the Boussinesq approximation to Eq. (3-47) above gives

Equation 3-48.

Prandtl mixing length hypothesis

Prandtl argued that the eddy viscosity could be expressed as the product of a turbulence length, lt,and a velocity scale, Vt, i.e.

Equation 3-49.

and that



Turbulence models

Equation 3-50.

Equation 3-51.

where κ is the von Karman constant. Combining Eqs. (3-49) to (3-51) gives

Equation 3-52.

Substituting Eq.(3-52) into (3-48) yields

Equation 3-53.

Rearranging gives

Equation 3-54.

The integration of Eq. (3-54) yields the famous log-law. However, before that, several other usefulrelations can be found.

Substituting Eq. (3-54) into (3-52) gives

Equation 3-55.

The production of turbulence kinetic energy, k, is

Equation 3-56.

So using Eq. (3-54) again

Equation 3-57.


Turbulence models


Local equilibrium assumption

In a turbulent boundary layer satisfying assumptions (3-47) to (3-50), the largest terms in theturbulence kinetic energy equation are the production and dissipation terms, which tend to be of similarmagnitude, but of opposite sign. So a further assumption is made that production equals dissipation

Pk=ρε

Equation 3-58.

so that

Equation 3-59.

Launder and Spalding's eddy viscosity

In the k-ε model, it is assumed that

μt=Cμk2/ε

Equation 3-60.

where Cμ is a constant. This is a general expression for μt valid everywhere, including in the near-wallregion. Equating Eqs. (3-60) and (3-55) gives

Equation 3-61.

Substitute Eq.(3-59) for ε and simplify

Equation 3-62.

Note that the equations for μt, Pk, ε and k (Eqs. (3-55), (3-57), (3-59) and (3-62) respectively)were all derived without recourse to the log-law, they do not involve any unresolved velocity scalesand they are all explicit in terms of τ/ρ.

The actual logarithmic relation follows from Eq. (3-54). Integrating it gives

Equation 3-63.



Turbulence models

where y* is a length scale and E is a dimensionless constant (E/y* is the constant of integration).This can be rearranged to

Equation 3-64.where

Equation 3-65.so that a velocity scale, u*, is used instead of the length scale.

Once a velocity scale is selected, the constants K and E in Eq. (3-64) can be matched withexperimental data. Traditionally, and quite reasonably, the choice for the velocity scale has been

Equation 3-66.With this value, constants of K = 0.41 and E = 8.43 correlate well to a range of boundary layer flowsfor smooth walls, usually to the degree that the assumptions (3-47) to (3-52) hold. In other words, ifvelocity measurements, U, where made for a large number of flows, then the dimensionless velocity u+/- = u/u* would approximately follow the same logarithmic prole:

Equation 3-67.These relations can be used to compute a wall shear without resolving the details of the near-wallregion. The algorithm proceeds by taking a near-wall velocity (computed from the Finite Volumeequations for the conservation of momentum) and then solving for the other dependent variables.

Further Considerations for the k-ε Model

The near wall velocity gradient can be written as:

Equation 3-68.which, using Equation (3-66) becomes

Equation 3-69.


Turbulence models


The near wall production used in the k-ε is obtained by combining Equations (3-57) and (3-69),which gives

Equation 3-70.

The definition for the velocity scale u* given by Eq. (3-66) is not suitable for regions of flow separation,reattachment and stagnation points where τω goes to zero. For the k-ε model, an alternative is tomake use of 3-62 as follows:

Equation 3-71.

From Eq. (3-70) and (3-69), combined with the assumption of equilibrium (Eq. 3-57), the nearwall dissipation is given by

Equation 3-72.

The factor f is a damping factor used to model the influence of the wall on the dissipation level.It is defined as [17]:

Equation 3-73.

This influence of f on the near wall dissipation is significant only if the near wall node is at y+<5.



Chapter 4: Source terms

Buoyancy force

Buoyancy force

In the presence of buoyancy, the gravity force is included in the source term SUj of Eq. (2-2), i.e.

Equation 4-1.

However, incorporating this gravity force as such into the source term can lead to round-off problems.To avoid such problems, the gravity force is implemented as follows:

The density is first expressed with respect to a "reference density", ρr, as

Equation 4-2.

then, Eq. (4-1) is re-written as

Equation 4-3.

or

Equation 4-4.

with



Equation 4-5.

P* is the pressure field with respect to the hydrostatic variation, i.e.

Equation 4-6.

and Po is an “offset" pressure (the real pressure when z = 0). It is the pressure P* ((-2/3)ρkδij ; seeSection Turbulence models) that is computed in the code, and pressure boundary conditions arein terms of P*.

The term (ρ— ρr)gj in Eq. (4-4) is the buoyancy force per unit volume. It is modeled as follows:

Take the differential of density to give

Equation 4-7.

Assume that the fluid density is not a function of pressure (e.g. an incompressible liquid).

Then,

Equation 4-8.

where β is the coefficient of thermal expansion defined as:

Equation 4-9.

Then we can say

Equation 4-10.

or

Equation 4-11.



Source terms

where Tr is a reference temperature, at the same condition as ρr. Thefefore,

Equation 4-12.

For an ideal gas, β is evaluated at the Tr condition. The expression is then exact:

Equation 4-13.

when β = 1/Tr use P/R=Tρ to obtain:

(ρ-ρr)=-ρβ(T-Tr)

Equation 4-14.

Note that this assumes that pressure-density effects are negligible.

The above reference temperature, Tr, is defined in two different ways in the code. Which one is used,depends on the boundary conditions applied to the model.

Enclosed cavities

In an enclosed cavity, with no communication of pressure level to the outside, Tr is truly arbitrary:the temperature and velocity fields are independent of it, and pressure reacts with differing linearvariations superimposed on an unchanging field. In effect, different Tr values only create differenthydrostatic pressure variations in P*.

Therefore, Tr can be selected by any other criterion; it is selected such that the total buoyancyforce summed over the whole domain is zero.

The buoyancy source term is

Equation 4-15.

This code reference temperature is computed internally to minimize the summed effect of thebuoyancy term,

Equation 4-16.


Source terms


where v is the volume, and the sums are over the whole domain. Substituting this expression for Trinto Eq. (4-15) and then summing the result over the whole domain gives a zero buoyancy force.

Cavities with openings

Consider a simple cavity or vent, open to a large domain at the top and bottom, and heated fromits internal walls. Physically, natural convection results with fluid entering through the bottom andheated fluid leaving through the top.

With Eq. (4-15) as the buoyancy source term, if Tr is computed from Eq. (4-16) then, by design, therewill be a zero total buoyancy force and with the zero force from the P* pressure boundary conditionsthere will be no net movement of the fluid up through the cavity. Because of this potential problem,the reference temperature for open cavities is not calculated from Eq (4-16); instead, is it simply takenas the maximum of all the vent temperatures in the enclosure, i.e.

Tr = Max(Tvent)

Equation 4-17.

Centripetal and Coriolis forcesFor flows in a rotating frame of reference (RFR), additional forces are introduced in the momentumequations, Eq. (2-2), due to the rotation. These are the centripetal and the Coriolis forces. In vectorform, the source term in the momentum equations is expressed as

Equation 4-18.

where is the rotational velocity vector (rad/sec), , is the velocity

vector, , and is the location vector with respect to the axis of rotation,

If , we can write:

Equation 4-19.

The components of the location vector are computed using Rx = x-xo; Ry = y-yo; Rz = z-zo where (xo,yo, zo) are the coordinates of the origin of the rotating frame of reference.

To obtain the components of the rotational velocity, the orientation of the axis of rotation is required.The axis of rotation is defined by its end points. The convention used here is to take the origin of theRFR as the origin of the axis and to define point B as the end point. One can define:



Source terms

Rwx=xB-xo; Rwy=yB-yo; Rwz=zB-zoEquation 4-20.

then

Equation 4-21.

Note that the direction of the rotation depends on the position of point B with respect to the originof the RFR.

It is important to note that the velocity on a RFR is relative to the rotating frame. If the model has afixed frame and a rotating frame, the velocities are relative to their respective frame of reference.

Flow resistance through porous materials

Flow resistance through porous materials

The pore structure resists the flow of the fluid passing through the porous material.

• An isotropic porous material has the same loss coefficient in all directions.• An orthotropic porous material has three different loss coefficients that correspond to the three

orthogonal principal axes.

The resistance to flow is included in the source term of the momentum Equation 2-2 as describedby the Darcy-Forchheimer law [26].

Isotropic resistance

For an isotropic porous material, the source term accounts for the resistance to flow as follows:

Equation 4-22.

where

• K is the permeability.• R is the inertial loss coefficient, and represents the fraction of the dynamic head lost per unit

distance and has a dimension of inverse length.

• is the magnitude of the velocity. The velocity is expressed in unit vectors of the global

coordinates as

Orthotropic resistance

An orthotropic material has three orthogonal principal axes. Each axis has a different loss coefficient.


Source terms


The components of the resultant resistance force per unit volume along the principal axes X1, X2,X3 are

Equation 4-23.

where

• R11, R22, and R33 are the inertial loss coefficients in the directions of X1, X2, and X3 respectively.• K11, K22, and K33 are the permeabilities in the directions of X1, X2, and X3 respectively.• U1, U2, and U3 are the velocity components in these principal coordinate directions.

Equation 4-23 is written in matrix form as follows:

Equation 4-24.

The resistance force in the global Cartesian coordinates is:

Equation 4-25.

where [T] is the transformation matrix between the coordinates along the principal axes and theglobal Cartesian system.

The components of the velocity along the principal axes are expressed in the global Cartesianvelocity components as follows:

Equation 4-26.



Source terms

where superscript T represents the transpose of the matrix.

Equations 4-24, 4-25, and 4-26 are combined to obtain the resistance force in the global Cartesiancoordinate system:

Equation 4-27.

This is the source term for the momentum Equation 2-2 in the case of flow through porous materialswith orthotropic resistance. The resistance force is nonlinear and is re-computed at each iteration.

Heat sourcesSimcenter Flow permits transient analysis with a constant or time-dependent heat source on thefluid side. In such cases, the source term, Sh in Eqs. (2-7 and (2-8) ), corresponds to that heatgeneration per unit volume.

The heat source is imposed on selected fluid elements through the use of a generic entity. Thisfeature was developed to simulate arc faults but can be used for any flow situation where a heatsource on the fluid side is present.

Condensation and evaporation

Condensation and evaporation

In transient analysis, condensation and evaporation at walls can be modeled when the ambientfluid is air.

The model assumes film type condensation or evaporation with the following conditions:

• The water film on the surface of the walls is very thin.

• The temperature of the liquid film is assumed to be the same as that of the wall.

• The rate of mass transfer between condensate and the air is small.

• The presence of the condensate does not affect the heat transfer coefficient of the surface.

Mass and scalar conservation equations

The flux of water vapor from the film to the air is modeled as [20]

Equation 4-28.


Source terms


If the water vapor density is greater than the density at saturation (i.e. 100% relative humidity), thewater vapor flux is then evaluated from

Equation 4-29.

and the mass transfer coefficient, hm, is evaluated from the Lewis relation i.e.:

Equation 4-30.

Where

• ρBsat(Ts) is the saturation density of water vapor at the wall temperature.

• ρBf is the density of the water vapor in the fluid.

• h represents the heat transfer coefficient.

• Cpm is the specific heat per unit volume of the air-vapor fluid mixture

• ρAf is the density of air in the fluid

• α/Dν is the Lewis number (ratio of the thermal diffusivity of the fluid mixture to the diffusivityof the water vapor).

Evaporation corresponds to a positive flux of water vapor to the fluid, while condensation correspondsto a negative flux.

This mass flux is a source term in both the mass conservation equation Eq. (2-1) and the scalarconservation equation Eq. (2-9).

Energy conservation equation

The addition or retrieval of water vapor to or from the air as a result of evaporation or condensationalso affects the overall enthalpy of the fluid mixture. The energy flux to the fluid mixture resulting fromevaporation or condensation is evaluated as

Equation 4-31.

where Cp,bf is the specific heat of the water vapor at the fluid temperature, Tf.

This energy flux is a source term in the energy equation Eq (2-2)



Chapter 5: Boundary conditions

Walls

Solid walls and flow surfaces

This section presents the numerical modeling of walls and flow surfaces.

Mass equation

No fluid can flow through a solid wall, hence the mass flow specified at the wall is simply zero.

Momentum equation, κ, and ε equations

For the momentum equations and the κ and ε equations, the treatment at the wall depends onwhether a “slip” or “no-slip”, stationary, translating, or rotating wall boundary conditions is applied.

Slip wall

A slip wall boundary condition is used to simulate the flow next to a frictionless surface or at a far fieldboundary. The wall shear stress and the velocity gradients normal to the wall are specified as zero(i.e. the velocity of the fluid relative to the wall is non-zero). Furthermore, with the κ-ε turbulencemodel, the gradients of κ and ε normal to the wall are also set to zero.

No-slip wall

The velocity of the fluid at a no-slip wall is set equal to the velocity of the wall, i.e. for a stationarywall, the fluid velocity at the wall is zero, while for translating and rotating walls, it is non-zero. Thecalculation of the wall shear stress, τω, depends on whether the flow is laminar of turbulent:

For a laminar flow, τω, is calculated directly from

Equation 5-1.

where ∂V/∂n is the velocity gradient normal to the wall. Note that the mesh close to the wall for alaminar flow has to be relatively well refined in order to get a good approximation of the velocitygradient at the wall. For turbulent flows, it would be impractical to fully resolve the details of theboundary layer or near-wall region. For this reason, a semi-analytical approach is used wherebythe effects of the near-wall region are modeled by certain near-wall relations or “wall functions”.Essentially, they allow the computation of wall shear given certain flow information at some distancefrom the wall.

The derivation of the near-wall relations for turbulent flow depends on several assumptions. Theseare presented in section Assumptions.



Let y+, a dimensionless wall distance for a wall bounded flow, or dimensionless velocity, be

Equation 5-2.

where the shear velocityu*, is defined as

Equation 5-3.

and y is some distance from the wall.

In the log-law region, the near-wall relation is

Equation 5-4.

where uf if the fluid velocity at some distance y from the wall, and κ and E are constants. FromEqs. (5-3) and (5-4):

Equation 5-5.

The wall shear stress is evaluated through Eqs. (5-2) to (5-5). The near wall turbulent viscosity forthe fixed viscosity model is estimated in the same fashion as for the rest of the fluid domain, throughEquation (3-5). For the mixing length model, Equation (2.28) is used with the appropriate mixinglength l from Equations (3-7) and (3-8) and the strain rate from Equation (3-12). In the case of thek-ε model, the near wall turbulent viscosity is obtained from:

Equation 5-6.

where the turbulence length scale is given by:

Equation 5-7.

and fμ is the Van Driest damping factor (Eq. (3-8)(4-2)). When the κ-ε turbulence model is used, thenear wall transport equation for ε is replaced by (see section Assumptions for details)



Boundary conditions

Equation 5-8.

where fε is a near wall damping function given by [17]:

Equation 5-9.

The near wall production of turbulent kinetic energy is evaluated from

Equation 5-10.

Extensions of the momentum wall function

Very close to the wall in the viscous sub-layer, the flow profile departs from the log-law andapproaches a linear profile, so that:

Equation 5-11.

is computed as for laminar flow. Typically this departure starts at around y+= 30 and is completearound y+= 5. Some sort of smooth transitional or “buffer” region exists in between. It is necessary tomodel this behavior in the wall function, not so much for accuracy, but more for numerical robustness;considering that the logarithmic function returns completely incorrect negative values for u+ and y+values very close to the wall (and becomes singular when y+ = 0).

Note

Having computational nodes closer to the wall than the logarithmic region is inappropriateand will reduce accuracy. Using nodes to resolve the near-wall region requiresmodifications to other aspects of the turbulence model, e.g. including low-Reynoldsnumber model terms, and requires many nodes. The considerations given here are only toensure that should this happen, the code's robustness is not compromised.

An effective way to implement the desired behavior is to use exponential blending functions betweenthe two limiting functions. In what follows, a general form is chosen that can also be used forthermal wall function.


Boundary conditions


In addition to the linear relation in the viscous sub-layer, the wall function needs to be modified forroughness effects. The log-law region is the region where the average velocity of a turbulent flow at acertain point is proportional to the logarithm of the distance from that point to the wall.

For the momentum case, in the log-law region, roughness effects can be correlated as follows:

Equation 5-12.

Equation 5-13.

where yr is the equivalent sand grain roughness. This shifts the log-law, but does not affect the slope.A generalized wall function that permits modeling of these two effects, and is extendible to thethermal near wall region, can be defined as follows.

Defining a general function for θ+ as:

Equation 5-14.

This general function is used for the momentum case, smooth or rough, with the following definitions:

Equation 5-15.

Equation 5-16.

Equation 5-17.

Equation 5-18.



Boundary conditions

Equation 5-19.

κ=0.41 C=5.2

Equation 5-20.

Plots of this function for u+ for varying dimensionless roughness is shown in Figure 5-1. Forreference, the smooth wall log-law and the linear function are also plotted as solid lines.

Figure 5-1. Full momentum wall function for varying roughness heights (u+ vs log10(z+) )(□ k+= 0 ; ∆ k+ = 10 ; + k+ = 100 ; x k+ = 1000; * k+ = 10000)


Boundary conditions


Note

1. The constant C is related to the previous constant E by E = eκC

2. The function Γ determines the blending between the two different regions. For asmooth wall (i.e. k+ = 0), it blends equal parts log-law and linear-law at a y+ of 8. Thez+ variable is used (z+ = y+ for smooth walls) so that the switch-over point is correctfor rough walls. Physically, for very rough walls, it is observed that the log-law regionextends to somewhat closer than y+= k+ and then quickly the velocity drops to zerobefore y+ = 0. The definition of z+ comes from requiring that the function gives 90%log-law at y+ = k+/2.

3. The parameter R replaces y+ in the linear sub-layer also, to cause the rapid drop tozero for rough walls. In fact, there is no laminar sub-layer when the wall is very rough.

Energy equation

For the energy equation, a solid wall can either be specified as adiabatic (no heat transfer allowedacross the wall), or convecting. In the former case, a heat flux of zero is imposed at the wall, while inthe later case, either the wall temperature, the heat flux or the heat load is specified at the wall.

For laminar flows, the heat flux between the wall and the fluid, qw, is related to the wall and fluidtemperatures through

Equation 5-21.

where k is the thermal conductivity of the fluid and yf is the distance from the wall at which Tf isevaluated.

For turbulent flows, the temperature variation in the thermal boundary layer follows the general wallfunction of Eq. (5-8) for a large number of boundary layer flows.

A dimensionless temperature, T+, can be defined in terms of properties, wall heat flux, and thenear-wall velocity scale u*:

Equation 5-22.

The general function Eq. (5-14) describes the universal boundary layer profile for temperature withthe following definitions of its arguments:

Equation 5-23.



Boundary conditions

Equation 5-24.

Equation 5-25.

Equation 5-26.

C = (3.85 Pr1/3—1.3)2

κ = 0.4717

Equation 5-27.

where Pr is the Prandtl number. With these definitions, this becomes exactly the thermal wall functionsproposed by B.A. Kader [9], accurate right to the wall, and for a wide range of Prandtl numbers.

Notice that there is no explicit reference to roughness and it is assumed that roughness effects arecaptured implicitly by the wall when y+ is defined through u*. Roughness τω and near wall k willincrease so that u* will be increased suitably (with either definition).

The assumed applicability of this wall function can be used to create a boundary condition for theenergy equation in a manner similar to the approach used for momentum. Equation (5-22) isre-arranged to,

Equation 5-28.

Given a near wall fluid temperature, Tf, Eq. (5-26) can be used to compute qw which is substituteddirectly into the finite volume energy equation at the wall.

For implementation clarity and consistency with the momentum treatment, the following manipulationsare made.

First, using the definitions of y+, Eq. (5-26) becomes

Equation 5-29.

which can also be written as


Boundary conditions


qw=h(Tw—Tf)

Equation 5-30.

From Eq. (5-29) it is clear that the effect of the wall function is to amplify the laminar (i.e. molecular)thermal diffusion to the wall. Also, as expected, the amplification term, (y+/T+)Pr, tends to 0 as y+tends to 0.

An additional complication arises in the implementation because the dependent variable for theenergy equation is not temperature but enthalpy, h. In general the relationship between these twois arbitrary. However, Eq. (5-29) can be modified so that an approximate dependency of qw on his retained, while not changing the converged answer for qw. So, denoting the “new” solution withsuperscript “n” and the “old” solution by “o”, Eq. (5-30) is written as,

Equation 5-31.

Clearly, on convergence, the wall heat flux does not depend on enthalpy.

Thermal wall function for high speed flows and flows with viscous heating

The standard approach to compute the wall heat flux in Simcenter Flow uses the following:

qw=h(Tw—Tf)

Equation 5-32.

where Tw is the wall temperature and Tf the fluid temperature at the near wall node, and h is the heattransfer coefficient. For flows with significant viscous heating however (such as high speed flows), theheat flux is not dependent on the difference between Tw and Tf but rather between Tw and a so-calledadiabatic wall temperature. This adiabatic wall temperature depends on the local flow conditions andcan be derived from the thermal wall function. Denoting LS as low speed and HS as high speed, thegeneral thermal wall function can be written as:

Equation 5-33.

where

Equation 5-34.

Equation 5-35.



Boundary conditions

Equation 5-36.

Equation 5-37.

Equation 5-38.

Equation 5-39.

In the above equations:

1. T +LS is the standard thermal wall function for low speed flows.

2. T +HS is the correction to the standard thermal wall function which accounts for the contributionfrom the viscous heating on the temperature profile.

3. Pr is the laminar Prandtl number.

4. Prt is the turbulent Prandtl number.

5. yf is the normal distance from the wall to the near wall node.

6. u* is the shear velocity.

7. qw is the wall heat flux.

8. ρ is the fluid density.

9. μ is the fluid dynamic viscosity.

10. Uf is the fluid velocity at the near wall node.

11. UC is the fluid velocity at the intersection of the linear and logarithmic temperature profiles (forlow speed flow).

Combining equations (5-32), (5-35) and (5-37), we get:


Boundary conditions


Equation 5-40.which allows to express the wall heat flux as:

Equation 5-41.

The term of equation (5-41) represents the standard low speed flow wall heat fluxwhile the term u*T*HS/T+HS is the contribution to the wall heat flux due to viscous heating. qw ispositive when heat flows from the solid to the fluid. Equation (5-41) can be rewritten as:

Equation 5-42.where:

Equation 5-43.In the literature for high speed flows, T*f is defined as the recovery temperature or the adiabaticwall temperature and is usually expressed as:

Equation 5-44.where r is the recovery factor.

If yf is located in the laminar sublayer (small yf+ value), Equation (5-43) can be written as

Equation 5-45.which gives the standard recovery factor for the Couette flow equal to Pr.

As for the implementation in Simcenter Flow of the thermal wall function for high speed flow, theuse of Equations (5-42) and (5-43) allows to keep the standard formulation for the heat transfercoefficient given by:



Boundary conditions

Equation 5-46.

The effects of viscous heating are then accounted for by correcting the fluid temperature so thatT*f replaces Tf when the heat flux is computed.

Humidity and general scalar equation

When the humidity equation or the other general scalar equation is solved, the gradient normal to thewall of the mass ratio Φ is simply set to zero.

General heat transfer coefficient for natural convection

General heat transfer coefficient for natural convection

In order to improve Simcenter Flow predictions for natural convection in turbulent flows, a new hcorrelation derived from a general temperature wall function for natural convection is implemented.The temperature wall function of Yuan et al. [23], which is valid for turbulent natural convection alonga vertical plate in air, is modified to make it valid for any Prandtl number and for any surface inclination.

A complete treatment of turbulent natural convection boundary layers should also include theimplementation of a natural convection velocity wall function. Implementing such a wall functionhowever, has implications in the treatment of the production and the dissipation of the turbulencekinetic energy in the k-ε turbulence model. Only the temperature wall function for natural convectionis implemented.

Description of the temperature wall function for natural convection

Using temperature measurements for turbulent natural convection along a vertical plate in air, Yuan etal. [23] were able to correlate the temperature profiles for a wide range of Re numbers using newtemperature and velocity scales. The following definitions are used:

Equation 5-47.

where:

• uq is the velocity scale based on heat flux.

• Tq is the temperature scale based on wall heat flux.

• y is the normal distance from wall.

• T is the local mean temperature.

• g is the gravitational acceleration constant.

• qw is the wall heat flux.


Boundary conditions


• Tw is the wall temperature.

• is the coefficient of thermal expansion.

• =v/Pr is the thermal diffusivity.

• v is the kinematic viscosity.

• Pr is the Prandtl number.

• ρ is the fluid density.

• Cp is the specific heat at constant pressure.

The temperature wall function of Yuan et al. is given by:

T* = y* for y* < 1

Equation 5-48.

T* = 1 + 1.36 ln y* 0.135 ln2y* for 1 < y* < 100

Equation 5-49.

T = 4.4 for y* > 100

Equation 5-50.

The above temperature wall function is valid only for a vertical surface in air. To make the wallfunction more general, the effect of Pr as well as the influence of the surface angle with respectthe gravity vector must be taken into account.

Based on the work of Yuan et al., it can be shown that, at the outer edge of the boundary layer (y* >100) where T = To, the value of the T* is given by:

Equation 5-51.

where Ra is the Rayleigh number and Nu is the Nusselt number. Using the correlation of Tsujiand Nagano [22] for a vertical at plate in air:

Nu=CtRa1/3 with Ct=0.12

Equation 5-52.

into equation (5-51) gives, for Pr=0.71, which is consistent with equation (5-50). If it is assumed thatfor any Pr number and for any surface angle, the non-dimensional temperature profiles in a naturalconvection boundary layer are similar, it is possible to express equations (5-49) and (5-50) in amore general fashion, using



Boundary conditions

T* = at +bt lny* + ct ln2y* for 1 < y* < 100

Equation 5-53.

T* = (Pr/Ct3)1/4y* for y* >100

Equation 5-54.

The coefficients in equation (5-53) are obtained by satisfying the following conditions:

T* = 1 for y* = 100 (based on Eq. (5-48)

T* = (Pr/Ct3)1/4y* for y* >154 (based on Eq. (5-54)

(based on Eq. (5-49)

This leads to:

Equation 5-55.

For Pr = 0.71 and Ct = 0.12, equations (5-53) and (5-54) become:

T* = 1 + 1.40 ln y* – 0.139 ln2y* for 1 < y* < 100

T = 4.5 for y > 100

Equation 5-56.

which agrees with the equations of Yuan et al. (equations (5-49) and (5-50)).

Several values of the coefficient Ct can be found in the literature. In general, it is observed that Ct isdependant on the Pr number as well as on the angle of surface with respect to the gravity vector.Raithby et al. [24] proposed the following general formulation:

Ct = [ Cu * (cos1/3(phi),0)max, Cv * sin1/3(phi)]maxEquation 5-57.

with the following definitions for the angle phi:

For Tw > To• phi=0 if the surface is horizontal, facing upward

• phi=180 if the surface is horizontal, facing downward

for Tw < To• phi=0 if the surface is horizontal, facing downward

• phi=180 if the surface is horizontal, facing upward


Boundary conditions


There is a certain degree of uncertainty in the determination of the Cu and Cv coefficients. Incroperaet al. [25] recommends a value of 0.15 for Cu while Raithby et al. [24] proposed a correlation whichaccounts for an influence of the Pr number. In Simcenter Flow, the correlation of Raithby is adoptedwith a small modification to make it compatible with the value of 0.15 for air. The Cu correlationadopted is:

Equation 5-58.

As to the Cv coefficient, there are significant discrepancies between the correlation proposed byRaithby et al. [24] and what is found in other heat transfer handbooks. The Raithby relationship forCv is:

Equation 5-59.

For Simcenter Flow, a new correlation for Cv as a function of Pr is derived so that the Nusseltrelationship Nu = CvRa1/3 matches as close as possible the more complex relationship recommendedby Incropera for vertical at plates which is:

Equation 5-60.

Figure 5-2 compares eq. (5-60) to the simple Nu relationship, for three different Pr numbers. The Cvcoefficient is adjusted for each Pr number in order to minimize the error with eq. (5-60).



Boundary conditions

Figure 5-2. Nu versus Ra for a vertical flatplate

eq. (5-60), Pr=0.71

eq. (5-60), Pr=10

eq. (5-60), Pr=1000

0.1145 Ra1/3

0.1460 Ra1/3

0.1590 Ra1/3

Although only three Pr numbers are illustrated here, a total of eight values were used ranging fromPr = 0.1 to Pr = 2000 with, on average, a difference of less than 4% between eq. (5-60) and thesimple Nu correlation.

Based on these results, a new Cv correlation is obtained which is:

Equation 5-61.

Figure 5-3 illustrates the differences between equations (5-59) and (5-61).


Boundary conditions


Figure 5-3. Variation of Cv with Pr

Cv estimated from eq. (5-60)

eq. (5-61)

eq. (5-59)

Summary of the equations used for the thermal wall function in natural convection

T* = y* for y* < 1

T* = at +bt lny* + ct ln2y* for 1 < y* < 100

T* = (Pr/Ct3)1/4y* for y* >100

with the following definitions

Ct = [ Cu * (cos1/3(phi),0)max, Cv * sin1/3(phi)]max

Getting the h correlation from the natural convection thermal wall function

The local heat transfer coefficient in Simcenter Flow is defines as:



Boundary conditions

Equation 5-62.

where Tf is the local fluid temperature at a normal distance from the wall yf .

Equation 5-63.

Combining equations (5-62) and (5-63), the local heat transfer coefficient is expressed as:

Equation 5-64.

One should note that, for laminar flow, y* < 1. In that case, equation (5-64) becomes, after somemanipulations,

Equation 5-65.

which corresponds to the standard heat transfer coefficient for laminar flows.

Based on the local wall temperature and heat flux on the surface of a wall element, as well as on thesurface orientation angle, the heat transfer coefficient is computed from equation (2.133) and thenatural convection thermal wall function.

Special considerations when calculating h

There are uncertainties regarding the validity of the correlations found in the literature when theorientation of the solid surface is such that phi > 150 degrees. In addition, as the angle phiapproaches 180 degrees (heated surface facing downward or cooled surface facing upward), laminarflow prevails even for very high Ra numbers. In Simcenter Flow, the methodology adopted for 150 <phi< 180 is to evaluate laminar and turbulent heat fluxes and select the highest one. This can bedone in a very straightforward fashion, as described next.

The heat flux for laminar flow can be written as:

Equation 5-66.

The near wall fluid temperature for laminar flow (Tf,lam) is expressed in terms of the nodal temperatureTn using:


Boundary conditions


Equation 5-67.

so that:

Equation 5-68.

For turbulent flow, the heat flux is:

Equation 5-69.

The near wall fluid temperature for turbulent ow (Tf,turb) is expressed in terms of the nodal temperatureTN using:

Equation 5-70.

which gives:

Equation 5-71.

It can be shown, after several manipulations of equation (5-71) that the turbulent heat fluxcorresponds to an amplification of the laminar heat flux, that is:

Equation 5-72.

The amplification factor tends to 1 when yf < 1 (laminar flow). Test done with SimcenterFlow indicate that the turbulent heat flux tends naturally towards the laminar heat flux for cases wherephi > 150 (heated surface facing downward or cooled surface facing upward).



Boundary conditions

Fans

Fans

The different fan types supported by the software are:

• Inlet fans

• Outlet fans

• Internal fans

• Recirculation fans

For each of these types, the software can either specify the flow rate through the fan (volume flowrate, mass flow rate or velocity), or calculate the fan's flow rate from a fan curve definition. Dependingon the fan type, you can also specify the following options:

• Pressure rise

• Flow angle

• Swirl

• Head loss

• Inlet temperature

• Temperature change from extract for recirculation fans

• Inlet pressure

• Inlet turbulent intensity and length scale

• Heat generation

Fan curve

Fan curve

A fan curve expresses the relationship between the pressure rise, Δp, through a given fan andthe volume flow rate, , circulated by that fan.

Equation 5-73.

In this equation, ν2 is the head loss coefficient. It appears if the fan's pressure rise reduces due to ahead loss. The software calculates this variable using Eq. (5-93).

A typical fan curve is illustrated in Figure 5-4.


Boundary conditions


Figure 5-4.

Fan's static pressure

Fan's volume flow rate

Fan curve

System curve

Fan's operating point

There are two different approaches to define the fan's volume flow rate from the fan curve:

Implicit approach In the implicit approach, the solver implicitly calculates the volume flow rateat current iteration (n+1) based on the pressure rise at the same iteration(n+1) and active coefficients of the linear system. The solver obtains the linearsystem coefficients from the fan curve derivative based on the fan's pressurerise at previous iteration (n). Therefore, both fan's pressure rise and fan'svolume flow rate are the results of the linear system solution.

Explicit approach In the explicit approach, the solver determines the volume flow rate at currentiteration (n+1) directly from the fan curve itself using the fan's pressure riseat previous iteration (n), and imposes the volume flow rate as an explicitboundary condition to form the linear system. The software then obtains thepressure rise at current iteration (n+1) by solving the linear system.

In the explicit approach, the pressure rise and volume flow rate must come to agreement throughexplicit iterations. This could lead to strong fluctuations in the solution convergence based onthe characteristics of the fan curve. You have to control these fluctuations using the relaxationtechniques. The implicit approach suppresses these fluctuations by making the pressure rise andvolume flow rate more in agreement at each iteration, through implicit coupling in the linear system.

The parallel flow solver supports both implicit and explicit approaches to define the volume flow ratefor all fan curve types; while, the serial flow solver only supports the implicit approach.

The parallel flow solver uses the implicit approach, by default. To switch to the explicit approach,you need to set the value of the EXPLICIT_FAN_CURVE advanced parameter to TRUE in a user.prmfile and save it in the run directory.

Implicit approach

Volume flow rate calculation with implicit approach

Using the implicit approach, the solver defines the relationship between the fan’s pressure rise andvolume flow rate using the Taylor series expansion.



Boundary conditions

Equation 5-74.

Where:

• f(Δp) represents the fan curve.

• The superscripts n and n+1 refer to the previous and current iteration levels, respectively.

The pressure rise, Δpn, volume flow rate, , and gradient, , are evaluated fromthe solution at iteration level, n. The updated volume flow rate, , is evaluated implicitly at thecurrent iteration level, n+1.

In the implicit approach, the solver takes the fan curve data local to elements over the two faces onopposite sides of the fan and obtains a normal velocity vector for each element of a face to impose

the Dirichlet boundary condition. The solver evaluates on the fan curve for a particular face

element and obtains from the derivative of the fan curve at the specified operatingpoint. In this approach, the solver inserts the local velocity into the solution matrix at a particularpoint based on the pressure difference across a face element. This gives a relationship betweenthe velocity and pressure.

Explicit approach

Volume flow rate calculation with explicit approach

The explicit approach consists of calculating the fan's pressure rise at a given iteration level using Eq.(5-76), and then obtaining the corresponding volume flow rate from the fan curve represented byEq. (5-75).

Equation 5-75.

The superscripts n and n+1 refer to the previous and current iteration levels, respectively. Thepressure rise, Δpn and are evaluated at iteration level, n. The updated volume flow rate, , isevaluated at the current iteration level, n+1. This volume flow rate is then imposed as the boundarycondition for the next iteration and a new value of pressure rise is calculated. However, when usingthe explicit approach, the pressure rise and the volume flow rate at a given iteration level are out ofsync until the solution starts to reach some level of convergence, which can lead to some fluctuationsin the solution convergence.

The solver then uses the calculated volume flow rate through the fan to obtain the normal velocityvectors on both faces of the fan: s1 and s2. The normal velocity vectors, and , becomeDirichlet boundary conditions on the opposing sides.


Boundary conditions


Pressure rise

Pressure rise

The fan's pressure rise is defined as the difference between the static pressure at the fan exhaust,PEs, and the total pressure at the fan intake, PIT.

Equation 5-76.

The solver calculates the average pressure values on the two sides of the fan by Eqs. (5-77) and(5-78).

Equation 5-77.

Equation 5-78.

Where f(s1) and f(s2) define the pressure distribution on both sides of the fan: s1 and s2.

The fan's pressure change is obtained as the difference between the average pressures on oppositesides of the fan.

Equation 5-79.

The fan's pressure rise is then modified based on the fan curve type which is discussed in detailin the next sections.

Pressure rise calculation for inlet fan curve

The fan's pressure rise is defined by Eq. (5-76).

The total intake pressure of an inlet fan is equal to the ambient pressure. Therefore, ΔP is actuallythe difference between the static pressure at the fan exhaust, PEs, and the ambient pressure, Pamb.The ambient pressure is the pressure at far field where stagnation condition applies.

Equation 5-80.

Considering the full gravitation momentum source, a new pressure rise is obtained by Eq. (5-81).

Equation 5-81.



Boundary conditions

The software calculates using Eq. (5-79). Having the fan's modified pressure rise, ΔP', the solverobtains the corresponding fan's volume flow rate using the fan curve.

Equation 5-82.

Pressure rise calculation for outlet fan curve


Considering the average hydrostatic contribution, a new pressure rise is calculated using Eq. (5-83).

Equation 5-83.

The software obtains the pressure values at the intake of an outlet fan from the static pressurevalues on the interior nodes. Therefore, to form the intake total pressure, the solver adds the intakedynamic pressure to the pressure rise equation.

Equation 5-84.

Now, the software uses the fan curve to get the outlet fan's volume flow rate.

Equation 5-85.

Pressure rise calculation for internal fan curve


The pressure values at the intake of an internal fan are obtained from the static pressure values onthe interior nodes. Therefore, to form the intake total pressure, the solver adds the intake dynamicpressure to the pressure rise equation.

Equation 5-86.

ρI=ρE and AI=AE because the fan's intake and exhaust collocates.

Now, the software uses the fan curve to calculate the fan's volume flow rate.

Equation 5-87.


Boundary conditions


Pressure rise calculation for recirculation fan curve


The pressure values at the intake of a recirculation fan are obtained from the static pressure valueson the interior nodes. Therefore, to form the intake total pressure, the solver adds the intake dynamicpressure to the pressure rise equation.

Equation 5-88.

Now, the software obtains the fan's volume flow rate from fan curve.

Equation 5-89.

Head loss on fan

In the case of a head loss, the fan curve is modified to reflect the fact that for a given volume flowrate, the pressure rise produced by the fan is reduced by the head loss. In the fan curve, the pressurerise is shifted below the nominal fan curve by the amount of ΔPloss.

Equation 5-90.

Where:

• CIs and CEs are the specified intake and exhaust loss coefficients, respectively.

• ρI and ρE are the intake and exhaust densities, respectively.

• V is the flow velocity.

Therefore, we can rewrite ΔPloss as follows:

Equation 5-91.

Where, AI and AI are the intake and exhaust areas.

Equation 5-92.

To obtain the volume flow rate, , from Eq. (5-73), the solver also needs the value for the headloss coefficient, ν2, that is defined in Eq. (5-93).



Boundary conditions

Equation 5-93.

CI and CE are the intake and exhaust loss coefficients, respectively, given by Eq. (5-94) and Eq.(5-95).

Equation 5-94.

Equation 5-95.

If there is no head loss on the fan, ν2 is set to zero.


Boundary conditions


Flow angle on fan

The flow through an inlet fan or an internal fan can be specified to be at an angle θ from thefan normal. In such cases, the velocity component normal to the fan, Vn, is the velocity obtainedfrom the fan curve, velocity, mass flow, volume flow or pressure rise specification. The velocitycomponent parallel to the fan plane, Vt, is then calculated so that the total velocity vector is in thespecified flow direction, i.e.:

Vn = Vspec

Equation 5-96.

Vt = Vntan θ

Equation 5-97.



Boundary conditions

Swirl on fan

Swirl can also be imposed on inlet and internal fans by specifying an axis of rotation and an angleθ from the fan normal. At each computation point on the fan, the velocity is calculated such that ithas a component normal to the fan plane and a component tangential to that plane in the directionof rotation.

The velocity component normal to the fan plane, Vn, is obtained from the mass flow rate, velocity, fancurve or pressure rise specification, as usual, whereas the tangential component, Vt, is calculated from

Vt = Vntan θ

Equation 5-98.


Boundary conditions


Energy equation in fans

Specified temperature

The fluid temperature can be specified as a boundary value on inlet fans only. The temperatureimposed can either be the ambient temperature or a temperature specified by the user.

Temperature change from extract to return

For recirculation fans, the user can specify a temperature change from the extract side of the fan tothe return side of the fan, i.e.

∆T = Treturn — TextractEquation 5-99.

This is equivalent to specifying the value of the return temperature

Treturn = Textract + ∆TspecEquation 5-100.

Heat generation

Heat generation can be specified on internal fans as well as on recirculation fans. The amount ofheat generated by the fan can be expressed as

Equation 5-101.

where is the mass flow rate through the fan, Cp is the specific heat at constant pressure of thefluid and ΔT = Texhaust-Tintake for internal fans, ΔT = Treturn-Textract for recirculation fans. The heatgeneration condition is in fact applied as a temperature specification at the exhaust side or returnside of the fan.

Turbulence quantities on fans

The solver applies turbulence quantities only at inlet fans and on the return part of the recirculationfans. Depending on the turbulence model and the specified turbulence quantities, the followingequations are used to compute the turbulence kinetic energy, k, and the dissipation rate, ε, or thespecific dissipation rate, ω:

Equation 5-102.

Equation 5-103.



Boundary conditions

Equation 5-104.

Equation 5-105.

When you do not specify any turbulence quantities, the solver computes them using:

• A turbulence intensity, Ix, of 4%

• A eddy length scale, lt, computed as follows:

Equation 5-106.

Equation 5-107.

where:

• U0 is the mean flow velocity at the boundary condition.

• μ is the fluid dynamic viscosity.


• A is the surface area of the fan.

For pressure-driven flows, the velocity is computed as follows:

Equation 5-108.

where ΔP is the maximum pressure difference of the complete fluid domain.

Humidity and scalar equation

Specified relative humidity or specific humidity

The relative humidity or the specific humidity can be specified at inlet fans even though the transportequation for water vapor in air is in terms of the mass ratio. The conversions from relative humidity tomass ratio and from specific humidity to mass ratio are described in Section 2-


Boundary conditions


General scalar equation

An inlet value of the mass ratio may be specified at inlet fans.

OpeningsOpenings, also called vents, are pressure and temperature specified boundaries. The conditionimposed depends on the direction of the flow, i.e. if the flow is an inflow or an outflow. The directionof the flow at a vent can vary during the solve.

For the serial solver, each element of a vent can be an inflow vent or an outflow vent separatelyfrom its neighbors.

For the parallel solver, all elements of an opening are either on an inflow vent or on an outflowvent at the same time.

Inflow vent

Pressure At inflow vents, you specify the total pressure, Pspec = Ptotal, but the solverimposes the static pressure:

Equation 5-109.

where Vn is he velocity component normal to the vent.

You can also specify a head loss coefficient, f, at a vent to simulate a screenor a filter at that opening. In such cases, the static pressure is computed asfollows:

Equation 5-110.

Temperature The vent temperature is applied at inflows only. The temperature is either thespecified value or ambient temperature.



Boundary conditions

Turbulencequantities

The solver applies turbulence quantities only at inflow vents. Depending onthe turbulence model and the specified turbulence quantities, the followingequations are used to compute the turbulence kinetic energy, k, and thedissipation rate, ε, or the specific dissipation rate, ω:

When you do not specify any turbulence quantities, the solver computesthem using:

• A turbulence intensity, Ix, of 4%

• A eddy length scale, lt, computed as follows:

where:

• U0 is the mean flow velocity at the boundary condition.

• μ is the fluid dynamic viscosity.


• ΔP is the maximum pressure difference of the complete fluid domain.

• A is the surface area of the vent.

Relative humidity orspecific humidity

The value of the relative humidity or of the specific humidity can be specified atvents. This value is applied at inflows only.

Mass ratio forgeneral scalarequation

An inlet value of the mass ratio may be specified at vents. This value is appliedat inflows only.


Boundary conditions


Angle from vent An angle that is specified at a vent will affect inflows only. The velocitycomponent parallel to the vent, Vt, is calculated so that the total velocity vectoris in the specified flow direction, i.e.

Vt = Vntan θ

Equation 5-111.

where Vn is the velocity component normal to the vent and θ is the anglebetween the flow direction and the vent normal.

Outflow vent

At outflow vents, you can only specify the pressure. The solver imposes the static pressure thatyou specify:

Equation 5-112.

or, if the vent is to ambient

Equation 5-113.



Boundary conditions

Convective outflowThe convective outflow boundary condition models fluid exiting the fluid domain without specifyingthe pressure. It lets the flow field exit and enter the flow domain on each element of the boundaryas necessary conserving the mass.

Note

The convective outflow boundary condition is supported only by the parallel flow solver.

At the centroids of each face element lying on the convective outflow boundary, the dependentvariable fields, φ, are computed from a discretized form of the advection equation:

For the purposes of the above boundary condition equation, the advecting velocity field is taken asuniform, in the direction normal to the boundary. The solver computes the magnitude of the velocity,U, from the average velocity of all other flow boundaries to conserve mass of the flow domain.

To compute the diffusive and advective fluxes, the solver derives an expression for the values of thedependent variables at the boundary face elements as a function of the values at the nodes of theadjacent solid element, and imposes them implicitly.

For the mass equation on the convective outflow boundary, the outflow velocity field is computedexplicitly based upon the current nodal field. It is then scaled to conserve mass, accounting fordensity variation with time.

Transient initialization

Because the temporal derivative is often dominant in the boundary value equation, a reasonable initialcondition is crucial; to this end, a small number of initializing steps are performed in which the outflowboundary is treated as an exhaust opening, to obtain a reasonable and conservative starting point forthe solution of the evolution of the boundary field.

Steady state assumption

For better stability, in steady state runs, the advected values are computed from azero-normal-derivative condition, i.e. with temporal derivative explicitly neglected. This assumptionis true in the converged steady state solution. It removes the need for the initializing iterations andimproves convergence and robustness. Because the conservation of mass is explicitly enforcedover the entire domain, the pressure field at the outflow boundary can evolve naturally, so that theupstream flow features are not strongly influenced by nearness of the boundary.


Boundary conditions


Supersonic inletIn Simcenter Flow you can use a supersonic inlet boundary condition, where you specify the Machnumber, inlet pressure, and inlet turbulence values.

The Mach number is defined as M=V/c, where V is the local speed, and c is the speed of sound.

The speed of sound c is the speed at which sound propagates through a medium under specificconditions.

In gases, c is dependent on the molecular weight, and it is a function of temperature c=(kRT)0.5,where k is the adiabatic exponent, R is the gas constant, and T is the absolute temperature.

The Mach number you type, is converted in a velocity using the speed of sound of the fluid used.This velocity is then applied to the faces you select as an inlet fan.

The general energy equation the solver uses for all flows including high speed flow is:

Equation 5-114.

Flows at velocities above Mach 0.3 have density changes of more than 5% and generally are treatedas compressible.

For low speed flows (Mach < 0.3) the energy equation is simplified. The pressure work and dissipationterms are neglected, the equation becomes:

Equation 5-115.

Screens

Screens

The flow solver models planar resistances such as screens by calculating the pressure drop, Δp,across them:

Equation 5-116.

where:

• is the velocity component normal to the screen.

• f is the head loss coefficient.



Boundary conditions

The direction of the velocity components on either side of the screen are identical, unless a flow angleis specified at the screen. This flow angle and the pressure drop are accounted for in the massand momentum equations.

You must specify one of the following:

• The total head loss coefficient f

• The coefficient b, which is the linear proportionality constant that relates the pressure dropto the normal velocity component: Δp = bVn

In this case, b has units of mass flux.

• A correlation to calculate the head loss coefficient.

If you choose to use a correlation to calculate the head loss coefficient f, you must chooseone of the following:

o Thin perforated plate screen correlation

o Wire screens correlation

o Silk thread screens correlation

Thin perforated plate screen correlation

For thin perforated plates the following definitions apply:

Where

• L is the length of the screen.

• W is the width of the screen.

• d is the diameter of a single orifice.

• P0 is the perimeter of a single orifice.

• A0 is the area of a single orifice.

• AFs =LW is the area of the free stream.

• FAR = (ΣA0)/AFs is the free area ratio.

• t is the thickness of the plate.

• dh=4A0/P0 is hydraulic diameter of a single orifice.

• ν is the viscosity of the fluid.

• Red = Vndh/ν is the orifice Reynolds number.


Boundary conditions


Figure 5-5. Perforated plate

For this geometry the following correlation applies:

f=(f'(1–FAR)0.5)+1–FAR)2/FAR2

Equation 5-117. Thin perforated plate screen correlation

Where the term f' depends on the geometry of the perforations from the screen.

To obtain f', you can select between the following models:

• Sharp Edge

• Rounded Edge

• Beveled Edge

Sharp Edge

f' = 0.707

The following restrictions apply:

• t/dh < 0.015

• Red > 105 where Red.

Figure 5-6. Sharp edge orifice



Boundary conditions

Rounded Edgef'=0.03+0.47x10—7.7r/dh


• Grid thickness is the same as orifice radius.

• Red > 103.

Figure 5-7. Rounded edge orifice

Beveled Edgef'=0.13+0.34x10f”

f''=-(3.4t/dh+88.4((t/dh)2.3)


• The beveled edge of the orifice is facing the flow.

• The bevel angle is between 40° and 90°.

• 0.01 < t/dh < 0.16

• Red > 104

Figure 5-8. Beveled edge orifice 40°-60°

Wire screens correlation

For wire screens the following correlation applies:

f=1.3(1–FAR)+((1/FAR)-1)2

Equation 5-118. Wire screen correlation

where:



Boundary conditions


• AFs is the area of the free stream.

• FAR = ΣA0/AFs is the free area ratio.

The following assumption applies: Red > 103

Silk thread screens correlation

For silk thread sreens the following correlation applies:

f=1.62[1.3(1–FAR)+((1/FAR)-1)2]

Equation 5-119. Silk thread screen correlation

where:


• AFs is the area of the free stream.

• FAR = ΣA0/AFs is the free area ratio.

The following assumption applies: Red > 500.

Symmetry boundariesA problem is symmetric about a plane when the flow on one side of the plane is a mirror image ofthe flow on the opposite side of the plane. In such case, the use of a symmetry plane conditionenables efficient use of computer resources by allowing the numerical solution to be obtained on afraction of the original domain.

The application of a symmetry plane condition to a planar surface of the grid means that the solutionon the complete geometry is a reflection of itself about the symmetry plane.

For the mass equation and all scalar equations (energy, water vapor or passive component, k and ε), a zero flow condition is imposed at the symmetry plane.

For the momentum equations, the symmetry condition is specified such that all vector values areparallel to the plane of symmetry.

Periodic boundariesPeriodic boundary conditions are used to simulate a flow leaving through a boundary A and enteringthrough a boundary B under identical conditions (velocity, temperature, scalar values, etc).

The periodic boundary conditions act as if the solution domain were rolled up so that boundaries Aand B were adjacent. Any pair of periodic boundaries must have similar shape and size, and caneither be parallel to one another, or not. In the later case, one boundary must be the copy of the otherperiodic boundary, rotated an angle phi with respect to a specified axis of rotation.



Boundary conditions

Non-parallel periodic boundaries can be used, for example, to simulate flow through a pipe withouthaving to model the entire 360 pipe cross-section: the user can model an angular section of the pipecross-section and impose periodic boundary conditions on the two artificial boundaries.


Boundary conditions

Chapter 6: Particle tracking

Particle tracking modelSimcenter Flow uses a lagrangian model and discrete particles. The equation of particle motion (6-1)is obtained by balancing the particle inertia and hydrodynamic forces.

Equation 6-1. Equation of particle motion

Where:

• W is the velocity of the particle

• U the velocity of the flow

• Cd is the drag of the particle.

• R is a random vector.

• ρp and ρL are the densities of the particle and the flow.

• vol and lelement are the volume and length of the element.

The different terms on the particle tracking governing equation represent in order:

• Particle inertia

• Buoyancy

• Pressure force

• Viscous drag

• Brownian diffusion

This formulation considers the effect of the fluid as if it were diverted by the particle with no wakeeffects included in the model.

Note

Brownian motion is the apparent random movement of particles suspended in a fluid dueto turbulence.



Turbulent particle motionThe turbulent particle interactions are modeled following a Brownian dynamic approach and usingthe turbulent viscosity μt obtained from the turbulence models.

Particle movement is estimated within the boundaries of an element during a particle time step.

A particle moves an average displacement |r| after a time scale t according to ‹|r|2› ≈ Dt, whereD is the particle diffusivity.

The estimated time of the particle within the element is used to determine the averaging time scale:

t ≈ 3√(volelement)/|U| ≈ lelement/|U|

Equation 6-2. Averaging time scale for particle tracking in turbulent flows

This particle diffusivity is given by

D=(kbT)/(6πμrp)

Equation 6-3. Particle tracking difussivity

Where:

• μ its viscosity

• kb is a constant of the fluid

• T its the temperature

• rp is the radius of the particles

The effect of turbulence in the particle is modeled adding the turbulent eddy diffusivity from theturbulence models into the particle difussivity as in Eq. (6-3).

D=(kbT)/(6πμrp) + (μt/ρ)

Equation 6-4. Particle tracking difussivity with turbulent effects



Chapter 7: Discretization

Control-volume methodThe governing partial differential equations described in 2- are discretized using a finite-elementbased control-volume method, [5] with this method, the governing equations are integrated over acontrol volume and over a time step. As an example taking the equation expressing the conservationof the passive component, the integration is expressed as:

Equation 7-1.

where V is the control volume, and δt is the time step. Using Gauss's theorem, Eq. (3.1) becomes:

Equation 7-2.

where nj is the unit outward surface normal of the surface of the control volume, and A is theouter surface area of the control volume. The volume and surface integrations are approximatednumerically over a discrete finite volume defined on a computational grid or mesh. For instance, thetransport, or advection, term is approximated as:

Equation 7-3.

where

Equation 7-4.

is the discrete mass flow through a finite sub-surface of the finite volume, ip denotes the integrationpoint of this sub-surface, ΔA is the sub-surface area, ip is the discrete value of Φ at this sub-surfaceand the sum is over all surfaces of the finite volume.



Element Node Element sector

Finite Volume Integration point

Figure 7-1.

In the element-based finite volume method, the finite volume is defined from elements, [5], asillustrated in 7-1. All the dependent variables, including pressure and the velocity components, arestored at the element nodes (i.e. it is a co-located method).

Each finite volume sub-surface is an element bi-sector plane, as shown in 7-2, where a complete finitevolume results from two surrounding quadrilateral elements and three triangular elements. Thesesub-surfaces are called integration point surfaces, and the integrand to be evaluated is computed attheir mid-points. This method of finite volume definition extends directly to 3D.

Figure 7-2.

Taking one element in isolation as shown in figure 7-2, the integration point surfaces and thefinite volume sectors, are indicated. The discretization scheme thus entails representing discreteapproximations to the volume integrals over the element sectors and discrete approximations to thesurface integrals on the element integration point surfaces. Robust, efficient and accurate schemesto achieve this have been implemented. They are detailed for each governing equation in sectionDetails of the discretization for the mass conservation equation. The overall equation assemblyproceeds by visiting each element in turn, making the discrete approximations to the terms in theintegrals, so that when all elements have been visited, every node has a completed finite volumeequation for each conservation law.



Discretization

Details of the discretization for the mass conservation equationUpon integration, the mass conservation equation, Eq. (2-1) gives

where the superscript o refers to the value at the previous time step, and Δnj = njΔA.

Equation 7-5.

For gases, the transient term is expressed as

Equation 7-6.

where R is the ideal gas constant.

The velocities at the integration point, Uj,ip are evaluated from momentum equations for theintegration point velocities, [5]. The resulting form of the mass conservation equation contain apressure redistribution term, which resolves the pressure-velocity coupling problems that is typicalof colocated methods, [6].

Details of the discretization for the other conservation equationsThe discretized form of the momentum conservation equation for velocity Uj , Eq. (2-2) can bewritten as

Equation 7-7.

where μeff = μ +μt.

Similarly, the discretized conservation equations for energy, water vapor, passive components,turbulence kinetic energy and dissipation rate are

Equation 7-8.


Discretization


Equation 7-9.

Equation 7-10.

Equation 7-11.

Equation 7-12.

where:

• keff = k +μt/Prt

• Dv,eff= Dv +μt/ρScv,t

• ρDΦ,eff = ρDΦ + μt/Sct

In Eqs. (7-8) to (7-12), the transient terms, convective terms, effective diffusion terms and sourceterms all have the same form from one equation to the other, and all are treated similarly in SimcenterFlow. The treatment of each of those terms and of the pressure term of the momentum equations isdescribed in the following sub-sections.

Transient terms

The transient terms of Eqs. (7-8) to (7-12) are implemented directly as such, and require no furthertreatment.

Convective terms

Convective terms

If we consider the discretized form of the passive component equation as an example, i.e. Eq. (7-10),

the term represents the summation of the convective (or advective) fluxes across theboundaries of a given control volume. The quantity Φip has to be evaluated in terms of nodal Φvalues. Various schemes can be used to evaluate this quantity. These schemes are known as firstorder or higher order schemes, based on the following simplified analysis:



Discretization

Consider the 1D representation of a control volume and its immediate neighbors illustrated in figure7-1. If we evaluate Φip or Φi+1/2 from a Taylor's Series, we have

Equation 7-13.

where H.O.T. stands for higher order terms. If Φi+1/2 is approximated by the first term of the Taylorseries only, then the truncation error is first order and the advection scheme is first order. Similarly, ifΦi+1/2 is approximated by the first two terms of the series, the truncation error is second order, andso is the advection scheme.

The default advection scheme in Simcenter Flow is first order. Four higher order schemes with fluxlimiters are also available.


Discretization


First order scheme

The first order scheme, upwind differencing scheme (UDS) or differencing scheme, Φip isapproximated by the value of Φ at the upstream node, i.e.

Φip=Φi.

Equation 7-14.

From Eq. (7-13), it is easy to determine that UDS is a first order scheme. With this scheme, thesolution is very stable and converges quickly. UDS can, however, give good results only if the flow isaligned with the mesh. If this condition is not fulfilled, there can be false diffusion in the solution field,which is characterized by serious smearing of expected gradients, [7]. In order to reduce the errorintroduced by false diffusion, higher order schemes have to be used.

Higher order bound schemes

With higher order bound schemes, or higher order schemes with flux limiters, various higher orderschemes can obtained from Eq. (7-13) depending on how the gradients in the Taylor series areevaluated.

Higher order schemes help reduce considerably the problem of false diusion, but they still have atruncation error. In the case of second order schemes, this truncation error is dispersive, while forthird order schemes, it is dissipative. In both cases, this may lead to unphysical oscillations andnumerical instability. Thus, higher order schemes as such give more accurate solutions than UDS,but they are, most of the time, less stable (oscillatory convergence), and are generally expensivein terms of calculation time.

In order to eliminate the oscillations that are inherent to most higher order schemes, it is possible toimpose a bound on the convected face values. This is the idea behind higher order bound schemes.The bounds are imposed through what are called “flux limiters”. Essentially, flux limiters, or limiterconstraints, are conditions that act to limit the face value, Φi+1/2, to lie between specified values. Theconcept of flux limiters is similar in many ways to the TVD schemes (Total Variation Diminishingschemes) that are widely used in gas dynamics [14].

Higher order schemes

Higher order schemes are available in Simcenter Flow. The recommended scheme is SOU (SecondOrder Upwind); this is also the default high order scheme. The other schemes available areUDS+NAC (Upstream scheme + Numerical Advection Correction), UDS+CDS (Upstream scheme +Central Dierence Scheme correction) and QUICK (Quadratic Interpolation scheme). The details ofthose schemes can be found in the literature, for example in [1-15, and 16].

All higher order schemes in Simcenter Flow are used with flux limiters to ensure boundedness. Insome cases such as turbulent flows being solved using the k-ε model, a solution is impossible toget without flux limiters. Details of the ux limiters used in Simcenter Flow are given in the followingsub-section.



Discretization

Flux limiters

In general, a limited higher order estimate for the value Φip of the dependent variable Φ at anintegration point whose upwind and downwind neighbors are nodes C and D respectively, may beexpressed according to the following equation

Φip = ΦC + βC,ip(Φip,HO - ΦC)

Equation 7-15.

where the subscript HO denotes the unlimited higher-order approximation, and βC,ip is the limiter(0 < βC,ip < 1). In this way, the limiter may be interpreted as a blending between the base UDSscheme and the higher order scheme, ideally selected to be the largest possible value whicheffectively prevents the development of undesirable artifacts, sush as unbounded values or spuriousoscillations, within the solution.

The limiters used for the SOU, QUICK, and CDS schemes in Simcenter Flow are based on theConvective Boundedness Criterion (CBC). The details of the implementation of this scheme are givenin [15] the following is a brief description of the CBC scheme.

Based on figure 1, the CBC condition requires the integration point value of Φ to be limited by:

ΦC < Φip < ΦD

Φip → ΦC as ΦC → ΦU or if ΦC is a local minimum or maximum

Equation 7-16.

The limiter is determined from the following equation

βC,ip= (Φip - ΦC)/(Φip,HO - ΦC)

Equation 7-17.

where Φip is chosen as the closest value to Φip,HO meeting the requirements of Equation 15.

When the HI-RES advection scheme is selected, the unlimited second-order NAC approximationbased upon the cell-centered gradient ∇ΦC at the upwind node is employed, according to thefollowing equation:

Φip,NAC = ΦC + (xip - xC)·∇ΦC

Equation 7-18.

The limiter is determined from the differentiable replaceable of the familiar Barth and Jespersenlimiter, proposed by V. Venkatakrishnan [27], according to the following equations

βC,ip = maxip(ΨC,ip)

ΨC,ip = (Δext(Δext + 2Δip) + Kδ3)/(Δ2ext + 2Δ2ip + ΔextΔip ± Kδ3)

Δext = Φext - ΦC

Δip = Φip,NAC - ΦC

Equation 7-19.


Discretization


where the maximization in the first equation of the Equation 19 is carried out over all intergrationsurfaces bounding the control volume about node C, δ is the average mesh spacing, and Φext isan extremal value over the stencil of node C.

If Δip > 0, then Φext is chosen as the maximum value over the stencil, otherwise, it is chosen as theminimum value, in either case to ensure boundedness of the reconstructed solution.

The factor K represents a blending of the limited and unlimited solutions, and determines themagnitude of oscillations in the solution which are effectively admitted as smooth by the limiter.

In Simcenter Flow, the value of K is computed based upon a scaling analysis of the parametersdefining the transport problem of interest, with the intent of effectively damping the growth of spuriousextrema while not unduly hindering convergence.

Diffusion termsThe diffusion terms in Eqs. (7-7) to (7-12) have the general form

Equation 7-20.

The derivatives at an integration point ip are evaluated by the using shape functions.

Shape functions are interpolation functions used to obtain values of field variables and derivativeswithin an element. Given an (s, t, u) coordinate within the element, the value of the field variable, forexample , Φ can be calculated using the shape functions, Nele:

Equation 7-21.

where NNODE is the number of nodes in the element.

Shape functions can be determined for an element, simply knowing (s, t, u) coordinates of theelement nodes, an appropriate polynomial in (s, t, u) and that the shape function corresponding to aparticular node is unity at that node and zero at all other element nodes.

The derivatives of the field variable, are then evaluated from Φ nodal values of and from thederivatives of the shape functions as follows:

Equation 7-22.



Discretization

Source termsThe discretized source terms are simply evaluated by multiplying Suj, Sh and S by the volume of thecontrol volume, as expressed in Eqs. (7-7) to (7-12).

Pressure term

The pressure term in Eq. (3.6) , , is evaluated using shape functions. The pressureat the integration point ip is determined from

Equation 7-23.

where Pnode are the nodal pressure values.

Rotating frames of referenceThe frozen disk methodology used for rotating frames of reference is useful for models where theflow distribution around the interface between the fixed and the rotating frames is non-uniform. Theinterface between the frames is called a RFR Interface and takes into account the frame change witha zero head loss across it. Flow recirculation is permitted across the interface.

Because there is no averaging across the interface, the flow solution in each frame is valid only forthe selected position of the rotating component.

7-3 below gives a schematic of the RFR Interface with coupled nodes and their control volumes.These nodes are physically at the same location but not in the same frame of reference. Node 1 ischosen to be in the RFR. The velocities at node 1 and 2 are related by

u1 = u2 - urot ; v1 = v2 - vrot ; w1 = w2 - wrot (or in vector form V1 = V2 - Vrot)

Equation 7-24.

while the static pressure is P1=P2. The rotation velocity components are given by

urot = wy*Rz - wz*Ry

vrot = wz*Rx - wx*Rz

wrot = wx*Ry - wy*Rx

where wx, wy, and wz are the components of the rotational velocity vector and Rx, Ry, and Rz are thecomponents of the location vector with respect to the origin of the RFR.


Discretization


Node 1Rotating Frame

Node 2Rotating Frame

Figure 7-3.

The equations for nodes 1 and 2 are first assembled considering all the fluxes except the onesthrough the RFR interface. Labelling the fluxes through the screens as Q1 and Q2, the control volumeequations for nodes 1 and 2 are written as

A1V1+B1=Q1

A2V2+B2=Q2

Equation 7-25.

where A1 and A1 are the coefficients resulting from the linearization of the equations and B1 and B2represent the connections to all the other nodes for each equations.

Summing the equations gives

A1V1+A2V2+B1+B2=Q1+Q2

Equation 7-26.

For the mass equation, Q1+Q2= 0. For the momentum equations, Q1=-mV1, Q2=-mV2 where m is themass flux. Hence Q1+Q2=m(V2—V1)

The mass and momentum equations for node 1 are obtained from combining Eqs. (7-24) and (7-26):

(A1+A2)V1+A2Vrot+B1+B2=Q1+Q2

Equation 7-27.

For node 2, the equations have the form

(A1+A2)V2+A1Vrot+B1+B2=Q1+Q2

Equation 7-28.



Discretization

Mixing PlaneA mixing plane boundary condition interfaces two or more fluid volumes with different flow conditionsin steady state problems.

The two interfacing faces:

• Can have different geometries.

• Can have a connected interface or be geometrically separated.

• Do not need to be parallel.

The algorithm solves each fluid volume independently. It uses velocity and pressure values from theadjacent fluid volume as boundary conditions.

The interfacing faces are subdivided in a number of areas you specify. The areas are matched to itsclosest corresponding area according to their geometry and the averaging method you choose.

Table 7-1. Averaging method examples

Along radiusAlong vector

Upstream

Downstream

Averaging direction

Number of segments (Ex. 5)

For each area, the pressures and velocities are transformed to a 1D field using an averagingtechnique and the information is exchanged to its corresponding area as follows:

• The averaged upstream velocity is applied to the downstream area as a velocity inlet.

• The averaged downstream pressure is applied to the upstream area as a pressure opening.

The values are mapped between areas to account for different geometry and mesh size at theinterfaces.

You define either a radial or axial averaging method.

• When the defined axis is perpendicular to the mixing plane, the velocity and the pressure areaveraged along radial arcs at the interfaces.


Discretization


• When the defined axis is parallel to the mixing plane, the velocity and the pressure are averagedalong axial segments at the interfaces.



Chapter 8: Solver numerical method

Solver numerical methodOnce the discrete finite volume equations have been assembled, they must be solved. Theimplementation assembles and solves the mass and momentum equations together, making it acoupled, as opposed to segregated, linear solver. This greatly improves robustness and efficiency.

The base iterative linear solver is a coupled Incomplete Lower Upper (ILU) factorization method. Asthis is not a direct solver, it is used within an iterative refinement loop, in which the exact solution isapproached with iteration. The system of equations can be written symbolically as:

A=xB

Equation 8-1.

where A is the coefficient matrix, x the solution vector (e.g. the U, V, W, P, H, k, ρv or Φ equation),and B the right hand side. Solving this iteratively, one starts with an approximate solution, xn, that isto be improved by a correction, x' to yield a better solution, xn+1, i.e.,

xn+1=xn+x'

Equation 8-2.

where x' is a solution of ,

Ax'=rn

Equation 8-3.

with r, the residual coming from,

rn=b-Axn

Equation 8-4.

The approximate iterative solver is used to solve the Ax'=rn. Repeated application of this algorithmwill yield a solution of the desired accuracy.

A particular implementation of algebraic Multigrid, called Additive Correction Multigrid is used, [12].This approach takes advantage of the fact that the discrete equations are representative of balancesof conserved quantities over a finite volume. Suitable merging of the original finite volumes to createlarger ones than create the coarser grid equations. This is done recursively, so that the hierarchy iscreated. The coarse grid equations thus impose conservation over a larger volume and by so doingreduce the error components at longer wavelengths. This is illustrated in Figure 8-1, where the nodesand finite volumes (not elements) are shown.



Figure 8-1.

In this hierarchy, current fine grid residuals are passed down to the next coarser grid, denoted by thedown arrows, with additional corrections to the relatively finer grid solution being obtained from thecoarser grid equations, denoted by the up arrows.

The ILU iterative solver improved by this Multigrid accelerator results in a complete linear solver thatdemonstrates no degradation of performance in the presence of large aspect ratios, and only alinear increase in cost with problem size. The current method employs a coupled algebraic Multigridaccelerator to prevent this behavior, [10].



Chapter 9: Temporal discretization

Temporal discretizationFor most of the transport equations, you can select either a fully-implicit (backward Euler) orsemi-implicit (Crank-Nicholson) scheme.

• The fully implicit scheme is first order accurate in time and unconditionally stable.

• The semi-implicit scheme is second order accurate with time step restrictions on stability.


Chapter 10: References

1. Currie, I.G. Fundamental Mechanics of Fluids, McGraw Hill, 1974.

2. Arpaci, V.S., Larsen, P.S. Convection Heat Transfer, Prentice Hall, 1984.

3. Rosehnow, Hartnett and Cho. Handbook of Heat Transfer, 3rd Edition, McGraw Hill.

4. Wilcox, DCW Industries. Turbulence Modeling for CFD, Second Edition, D.C.

5. Schneider, G.E. and Raw, M.J., Control Volume Finite-Element Method for Heat Transfer andFluid Flow using Colocated Variables- 1. Computational Procedure. Numerical Heat Transfer,Vol.11, pp.363-390, 1987.

6. Rhie, C.M. and Chow, W.L., A Numerical Study of the Turbulent Flow Past an Isolated Airfoil withTrailing Edge Separation, AIAA paper 82-0998, 1982.

7. Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington D.C., 1980.

8. Leonard et al., International Journal for Numerical Methods in Fluids, vol. 20, p. 421, 1995.

9. Kader, B.A.. "Temperature and Concentration Profiles in Fully turbulent Boundary Layers". Int. J.Heat Mass Trans., 21, N0.9, pp 1541-1544, 1981.

10. Raw, M.J., A Coupled Algebraic Multigrid Method for the 3D Navier-Stokes Equations. 10thGAMM Seminar, Kiel 1994.

11. Briggs W.L., A Multigrid Tutorial, SIAM, Philadelphia, 1987.

12. Hutchinson, B.R., Raithby, G.D: A Multigrid Method Based on the Additive Correction Strategy,Numerical Heat Transfer, Vol. 9, pp.511-537, 1986.

13. 1997 ASHRAE Fundamentals Handbook. Chapter 6: Psychometrics.

14. Sweeby, P.K., "High Resolution Schemes Using Flux-Limiters for Hyperbolic Conservation Laws",SIAM J. Numer. Anal., Vol. 21, pp. 995-1011, 1984.

15. Leonard, B.P., Drummond, J.E., "Why you should not use "hybrid", "Power-Law" or relatedexponential schemes for convective modeling - There are much better alternatives", InternationalJournal for Numerical Methods in Fluids, Vol. 20, pp.421-442, 1995.

16. Gaskell, P.H., Lau, A.K.C., "Curvature-Compensated Convective Transport: SMART, a NewBoundedness-Preserving Transport Algorithm", International Journal for Numerical Methodsin Fluids, Vol. 8, pp617-641, 1988.

17. Yap, C.R., “Turbulent Heat and Momentum Transfer in Recirculating and Impinging Flows”,Ph.D. Thesis, University of Manchester, 1987



18. Reynolds, W.C., Perkins, H.C.: Engineering Thermodynamics, McGraw Hill, 1977 .

19. Reid, R.C, Prausnitz, J.M, Poling, B.E. , “The Properties of Gases and Liquids”, McGraw Hill,fourth edition, 1986.

20. 1997 ASHRAE Fundamentals Handbook. SI Edition. Chapter 5.

21. R. Cheesewright, Turbulent natural convection from a vertical plane surface, J. Heat Transfer90, 1-8 (1968).

22. T. Tsuji and Y. Nagano, Velocity and temperature measurements in a natural convection boundarylayer along a vertical flat plate, Experimental Thermal and Fluid Science 2, 208-215 (1989).)

23. X. Yuan, A. Moser, P.Suter, Wall functions for numerical simulation of turbulent natural convectionalong a vertical plate, Int. J. Heat Mass Transfer 36, 4477-4485 (1993).

24. G.D. Raithby, K.G.T. Hollands, Natural Convection, Handbook of Heat Transfer, Third Edition,W.M. Rohsenow, J.P. Harnet, Y.I. Cho, McGraw Hill (1998).

25. F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, Fourth Edition, JohnWiley, Inc. (1996)

26. S. Whittaker, Volume Averaging of Transport Equations. In J. Prieur du Plessis (ed.). FluidTransport in Porous Media. Chap. 1. Computational Mechanics Publications: Southampton,UK (1997).

27. V. Venkatakrishnan, Convergence to Steady State Solutions of the Euler Equations onUnstructured Grids with Limiters. Journal of Computational Physics 118, 120-130 (1995).

28. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge, UK, 2002.

29. H. Schlichting and K. Gersten, Boundary Layer Theory, 8th edition, Springer, Berlin, 2000.

30. A. Einstein, Investigations on the Theory of the Brownian Movement, Dover, NY, 1956.

31. S.B. Pope, Turbulent Flows, Cambridge, UK, 2000.

32. G.R. Fowles and G.L. Cassiday, Analytical Mechanics, 6th edition, Brooks and Cole, Stamford,1999.



Index

CCondensation and Evaporation . . . . . . . . 4-7

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