SIEMENS
Simcenter Flow
Solver Reference
Manual
Simcenter • 11
Contents
Proprietary & Restricted Rights Notice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1
Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1
Mass and momentum equations . . . . . . . . .
Energy equation . . . . . . . . . . . . . . . . . . . .
Energy equation . . . . . . . . . . . . . . . . .
Low speed equation . . . . . . . . . . . . . .
High speed equation . . . . . . . . . . . . . .
Scalar equation . . . . . . . . . . . . . . . . . . . .
Scalar equation . . . . . . . . . . . . . . . . .
Modeling a passive scalar . . . . . . . . . .
Gas mixtures . . . . . . . . . . . . . . . . . . .
Humidity . . . . . . . . . . . . . . . . . . . . . .
Equations of state . . . . . . . . . . . . . . . . . .
Equation of state . . . . . . . . . . . . . . . .
Ideal gas law . . . . . . . . . . . . . . . . . . .
Redlich-Kwong real gas equation of state
Non-Newtonian fluid models . . . . . . . . .
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2-1
2-2
2-2
2-3
2-3
2-3
2-3
2-4
2-4
2-6
2-8
2-8
2-10
2-11
2-11
Turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1
Turbulence models . . . . . . . . . . . . . . . . . .
Fixed viscosity model . . . . . . . . . . . . . . . .
Mixing length turbulence model . . . . . . . . .
k-ε model . . . . . . . . . . . . . . . . . . . . . . . .
k-ω model . . . . . . . . . . . . . . . . . . . . . . . .
SST model . . . . . . . . . . . . . . . . . . . . . . .
Assumptions . . . . . . . . . . . . . . . . . . . . . .
Assumptions . . . . . . . . . . . . . . . . . . .
Prandtl mixing length hypothesis . . . . . .
Local equilibrium assumption . . . . . . . .
Launder and Spalding's eddy viscosity . .
Further Considerations for the k-ε Model
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3-1
3-2
3-2
3-3
3-4
3-5
3-8
3-8
3-8
3-10
3-10
3-11
Source terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1
Buoyancy force . . . . . . . . . . . . . . . . .
Buoyancy force . . . . . . . . . . . . . .
Enclosed cavities . . . . . . . . . . . . .
Cavities with openings . . . . . . . . . .
Centripetal and Coriolis forces . . . . . . .
Flow resistance through porous materials
Simcenter 11
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Simcenter Flow Solver Reference Manual
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4-1
4-1
4-3
4-4
4-4
4-5
3
Contents
Contents
Flow resistance through porous materials
Heat sources . . . . . . . . . . . . . . . . . . . . . .
Condensation and evaporation . . . . . . . . . .
Condensation and evaporation . . . . . . .
Mass and scalar conservation equations
Energy conservation equation . . . . . . .
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4-5
4-7
4-7
4-7
4-7
4-8
Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1
Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solid walls and flow surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Slip wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
No-slip wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Extensions of the momentum wall function . . . . . . . . . . . . . . . . . . . . .
Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thermal wall function for high speed flows and flows with viscous heating
General heat transfer coefficient for natural convection . . . . . . . . . . . . .
Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fan curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow angle on fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Swirl on fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Energy equation in fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Turbulence quantities on fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Humidity and scalar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convective outflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supersonic inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thin perforated plate screen correlation . . . . . . . . . . . . . . . . . . . . . . .
Wire screens correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Silk thread screens correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Symmetry boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Periodic boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5-1
5-1
5-1
5-1
5-3
5-6
5-8
5-11
5-19
5-19
5-19
5-26
5-27
5-28
5-28
5-29
5-30
5-33
5-34
5-34
5-34
5-35
5-37
5-38
5-38
5-38
Particle tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1
Lagrangian particle tracking . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . .
Equations of motion . . . . . . . . . . .
Modeling of the drag force . . . . . . .
Modeling of the perturbed flow field .
Modeling of chaotic motions . . . . . .
Boundary treatment . . . . . . . . . . .
Wall boundaries . . . . . . . . . . . . . .
Inlet and symmetry boundaries . . . .
Periodicity boundaries . . . . . . . . . .
Initial conditions for injected particles
Cunningham correction factor . . . . . . .
Sutherland's law . . . . . . . . . . . . . . . . .
4
Simcenter Flow Solver Reference Manual
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6-1
6-1
6-2
6-4
6-6
6-10
6-12
6-13
6-15
6-16
6-17
6-18
6-20
Simcenter 11
Contents
Particle tracking limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21
Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1
Control-volume method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Details of the discretization for the mass conservation equation .
Details of the discretization for the other conservation equations
Convective terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convective terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
First order scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Higher order bound schemes . . . . . . . . . . . . . . . . . . . . .
Higher order schemes . . . . . . . . . . . . . . . . . . . . . . . . . .
Flux limiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Diffusion terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Source terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pressure term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotating frames of reference . . . . . . . . . . . . . . . . . . . . . . . .
Mixing Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7-1
7-3
7-3
7-4
7-4
7-6
7-6
7-6
7-7
7-8
7-9
7-9
7-9
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
Simcenter 11
Simcenter Flow Solver Reference Manual
5
Proprietary & Restricted Rights Notice
© 2016 Siemens Product Lifecycle Management Software Inc. All Rights Reserved. This software
and related documentation are proprietary to Siemens Product Lifecycle Management Software Inc.
The Simcenter Flow solvers including the NIECE parallel flow solver and related documentation are
proprietary 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) Sandia
Corporation, is a free software licensed under the terms of version 2.1 of the GNU Lesser General
Public License as published by the Free Software Foundation. A copy of the license is found in
the[software_installation]\NXCAE_extras\tmg\install\GNU_LGPLicense.txt file of your distribution
and you can read it at http://www.gnu.org/licenses/lgpl.html. The Trilinos library may be obtained from
Sandia National Labs at http://trilinos.sandia.gov/. For a period of up to three years after the release
date of this software, licensed users can contact MAYA Heat Transfer Technologies LTD. to obtain the
source code of Trilinos used with this release, as well as the object code required to link NIECE with a
customized 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. Related
papers are available at http://www.cs.umn.edu/~metis, while the primary reference is “A Fast and
Highly 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 to
others. This software was authored by:
Mathematics and Computer Science Division
Argonne National Laboratory, Argonne IL 60439
(and)
Department of Computer Science
University of Illinois at Urbana-Champaign
GOVERNMENT LICENSE
Simcenter 11
Simcenter Flow Solver Reference Manual
7
Proprietary
& Restricted
Rights
Notice
Proprietary
& Restricted
Rights
Notice
Portions of this material resulted from work developed under a U.S. Government Contract and are
subject to the following license: the Government is granted for itself and others acting on its behalf a
paid-up, nonexclusive, irrevocable worldwide license in this computer software to reproduce, prepare
derivative works, and perform publicly and display publicly.
DISCLAIMER
This computer code material was prepared, in part, as an account of work sponsored by an agency
of the United States Government. Neither the United States, nor the University of Chicago, nor
any of their employees, makes any warranty express or implied, or assumes any legal liability or
responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or
process disclosed, or represents that its use would not infringe privately owned rights. Portions of
this 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 to
others. The code is licensed "as is." The User bears the risk of using it. Microsoft gives no express
warranties, guarantees or conditions. To the extent permitted by law, Microsoft excludes the implied
warranties 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 equations
for the conservation of mass, momentum, energy, and general scalars in general complex 3D
geometries. It uses an element-based finite volume method and a coupled algebraic Multigrid method
to 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, condensation
and evaporation at walls, and general boundary conditions for fluid flow and heat transfer. Details of
the mathematical model, the discretization of the equations, and of the solution method used in the
Simcenter 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 equations
The mass and momentum equations, when expressed in Cartesian coordinates and using the
tensorial 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 components
of the mean and the fluctuating velocity in the xj direction, P is the pressure, ρ is the density of
the fluid, μ is the dynamic viscosity of the fluid, and
represents the turbulent (or Reynolds)
are the source terms for the mass and momentum equations, respectively.
stresses. Sm and
Equation (2-1) expresses the conservation of mass of the fluid and is valid for incompressible and
compressible flows.
Equation (2-2) represents the conservation of momentum for general flows. The various terms of
this equation are in the same order of the equation: the transient term, the convection term, the
pressure 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.
Simcenter 11
Simcenter Flow Solver Reference Manual
2-1
Chapter
Governing
equations
Chapter
2: 2: Governing
equations
Energy equation
Energy equation
In order to describe completely deferent kinds of flows, the equation of conservation of energy
must also be implemented. Starting from total energy equation, the high speed form and the low
speed 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', a
heat 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 rate
of energy input per unit volume due to convection, the rate of energy addition due to conduction
and 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-off
purposes. For low speed incompressible and compressible flows, the pressure work and dissipation
terms in Eq. (2-6) can be neglected. The simplified form of the energy equation has the mechanical
energy 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 the
mechanical 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 more
general scalar equations that have the following form:
Simcenter 11
Simcenter Flow Solver Reference Manual
2-3
Chapter
Governing
equations
Chapter
2: 2: 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
Reynolds flux where Φ' is the fluctuating mass fraction of the scalar.
is the turbulent or
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 that
the 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 main
fluid, 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 two
gases are assumed to behave as perfect gases using the ideal gas law (see Ideal gas law for more
information).
Equations 2-1, 2-2, and 2-7or 2-8 represent the conservation of mass, momentum, and energy for
the gas mixture. The fluid properties calculated by the flow solver are the properties of the gas
mixture. In order to calculate these mixture properties, the user must supply the properties of the
second gas (scalar).
2-4
Simcenter Flow Solver Reference Manual
Simcenter 11
Governing equations
When modeling a gas mixture, the density, ρ, the specific heat at constant pressure, Cp, the thermal
conductivity, k, and the dynamic viscosity, μ of the gas mixture are calculated at every iteration. This
property update is based on the assumption that the two gases behave as perfect gases.
Equation 2-11. Ideal gas equation of state for Equation 2-12. Ideal gas equation of state for
gas 1
gas 2
where p1, p2, R1, and R2 represent the partial pressures of gas 1 and of gas 2, and the gas constants
of 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,
Property
, and gas 2,
Name
, the following table defines
and
.
Equation
The molar specific heat of the
mixture
Equation 2-16.
The molar 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 the
method of Wilke [19] which is valid for gases at low pressures. According to this method
Simcenter 11
Simcenter Flow Solver Reference Manual
2-5
Chapter
Governing
equations
Chapter
2: 2: 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, of
gas 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 the
conservation equations is obtained, the postprocessor coverts the mass ratio values back to relative
and specific humidity values.
Furthermore, the water vapor properties do not need to be supplied, as they are readily available in
the 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 the
saturation pressure of the water vapor at a given temperature, pv. sat(T):
2-6
Simcenter Flow Solver Reference Manual
Simcenter 11
Governing equations
Equation 2-23.
A value of φr equal to 100% indicates the onset of condensation. It can be seen from Equation 2-23
that the air/vapor mixture gets closer to the condensation state if water vapor is added to the fluid (the
partial 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
Value
5.6473590 x 10+3
9.6778430 x 10–3
2.0747825 x 10–9
Coefficient
C1
C3
C5
C7
Value
6.3925247
6.2215701 x 10–7
- 9.4840240 x 10–13
Coefficient
C2
C4
C6
4.1635019
For 0°C ≤ T< 200°C
ln(pv.
sat)
= C8/T + C9 + C10T + C11T2 + C12T3 + C13ln(T)
Equation 2-25.
with
Coefficient
C8
C10
C12
Value
- 5.8002206 x 10+3
- 4.8640239 x 10-2
- 1.4452093 x 10–8
Value
1.3914993
4.1764768 x 10–5
6.5459673
Coefficient
C9
C11
C13
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 2-24
and 2-25. Given the pressure of the water vapor/air mixture, p, the mass ratio is then obtained by
combining Equations 2-10 to 2-14 where the gas 1 properties are air properties and gas 2 properties
are water vapor properties:
Equation 2-26.
where:
Simcenter 11
Simcenter Flow Solver Reference Manual
2-7
Chapter
Governing
equations
Chapter
2: 2: 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 between
thermodynamic state variables for a given material. With the mass, momentum, and energy
equations, 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. These
material variables can vary with time, temperature, pressure, or they can vary with both temperature
and 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 interpolation
with 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.
2-8
Simcenter Flow Solver Reference Manual
Simcenter 11
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 flow
solver calculates the specific enthalpy, h, from the energy equation Eq. (2-6), and uses the following
equation 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 between
thermodynamic 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 the
following equation to calculate the temperature T:
h=cpT
Equation 2-30.
where cp is the specific heat at constant pressure of the gas.
Simcenter 11
Simcenter Flow Solver Reference Manual
2-9
Chapter
Governing
equations
Chapter
2: 2: 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).
2-10
Simcenter Flow Solver Reference Manual
Simcenter 11
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 and
valid 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 can
use one of the following models according to the behavior of the fluid:
•
Power-Law model
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Chapter
Governing
equations
Chapter
2: 2: 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 < μmax
Equation 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
•
μmax is the maximum viscosity limit.
min
is the minimum viscosity limit.
Herschel-Bulkley model
Equations 2-37 and 2-38 present how the fluid viscosity, μ, of a Herschel-Bulkley fluid is modelled in
Simcenter Flow.
Equation 2-37.
Equation 2-38.
where:
•
2-12
K is the consistency index.
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Governing equations
•
is the shear rate.
•
n is the power law index.
•
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 Newtonian
fluid 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 model
when 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.
•
is the shear rate.
•
n is the power law index.
•
T0 is the reference temperature.
•
T is the fluid temperature.
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Chapter
Governing
equations
Chapter
2: 2: Governing
equations
•
μ∞ is the infinite-shear viscosity.
•
μ0 is the zero-shear viscosity.
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Chapter 3: Turbulence models
Turbulence models
In Simcenter Flow, the flow field can be solved as laminar or as turbulent. Five turbulence models are
available 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,
are simply zero.
, and the Reynolds fluxes,
and
,
With all five models, the Reynolds stresses and fluxes are evaluated using a Boussinesq eddy
viscosity assumption [4] i.e.
Equation 3-1.
Equation 3-2.
Equation 3-3.
Equation 3-4.
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Chapter
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models
Chapter
3: 3: Turbulence
models
where k is the turbulence kinetic energy, Prt is a turbulent Prandtl number, and Sct , the turbulent
Schmidt number.
In the code, the right term in Eq. (3-1) ((-2/3)ρkδij) is included in the pressure term of the momentum
equation 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 you
simulate turbulent fluid flow.
Fixed viscosity model
For 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 is
taken as 1/7 of the cube root of the volume of the domain.
Mixing length turbulence model
The mixing length turbulence model is a zero-equation model which uses the following relationship
to 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:
Equation 3-7.
Equation 3-8.
κ is the Von Karman constant (κ =0.41), y is the normal distance from the node to the wall and ymax
is a characteristic length scale for the model. If this length scale is not specified by the user, a
default value is computed using
Equation 3-9.
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Turbulence models
where Vol is the volume of the fluid domain and Awetted is the wetted area. A length scale is computed
for each fluid domain in the model. Note that for a square or cylindrical duct, the above length
scale 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 rate
is given by:
Equation 3-10.
Equation 3-11.
For wall nodes, with u* as the shear velocity, the strain rate is based on the logarithmic wall function:
Equation 3-12.
k-ε model
With 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, Cμ
is 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 by
solving 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.
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Chapter
Turbulence
models
Chapter
3: 3: 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 model
With 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 kinetic
energy, and ρ is the density of the fluid.
The turbulent kinetic energy, k, and the specific dissipation rate of turbulent kinetic energy, ω, are
obtained 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:
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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 model
With 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:
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Chapter
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Chapter
3: 3: Turbulence
models
Equation 3-29.
Equation 3-30.
When F1 = 0, the transport equations are equivalent to the k-ε
transport equations are equivalent to the k-ω.
model, and when F1 = 1, the
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.
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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.
Equation 3-41.
Equation 3-42.
Equation 3-43.
Equation 3-44.
Equation 3-45.
Equation 3-46.
The following table lists the constants for this model.
Coefficient
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Value
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Turbulence
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Chapter
3: 3: Turbulence
models
0.09
1
2
σω1
σω2
2
0.075
0.44
2
1/0.856
1
σk1
σk2
a1
0.0828
5/9
2
1
0.3
Assumptions
Assumptions
The derivation of the near-wall relations for turbulent flows depends on several assumptions
presented 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 viscosity
and 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
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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 useful
relations 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.
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Chapter
3: 3: Turbulence
models
Local equilibrium assumption
In a turbulent boundary layer satisfying assumptions (3-47) to (3-50), the largest terms in the
turbulence kinetic energy equation are the production and dissipation terms, which tend to be of similar
magnitude, 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-wall
region. 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 scales
and they are all explicit in terms of τ/ρ.
The actual logarithmic relation follows from Eq. (3-54). Integrating it gives
Equation 3-63.
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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 with
experimental 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 flows
for smooth walls, usually to the degree that the assumptions (3-47) to (3-52) hold. In other words, if
velocity 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-wall
region. The algorithm proceeds by taking a near-wall velocity (computed from the Finite Volume
equations 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.
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Chapter
Turbulence
models
Chapter
3: 3: 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 to
make 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 near
wall 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.
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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
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Chapter
Source
Chapter
4: 4: Source
termsterms
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 ; see
Section Turbulence models) that is computed in the code, and pressure boundary conditions are
in 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.
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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 linear
variations superimposed on an unchanging field. In effect, different Tr values only create different
hydrostatic pressure variations in P*.
Therefore, Tr can be selected by any other criterion; it is selected such that the total buoyancy
force 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 the
buoyancy term,
Equation 4-16.
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Chapter
Source
Chapter
4: 4: Source
termsterms
where v is the volume, and the sums are over the whole domain. Substituting this expression for Tr
into 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 from
its internal walls. Physically, natural convection results with fluid entering through the bottom and
heated 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, there
will be a zero total buoyancy force and with the zero force from the P* pressure boundary conditions
there 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 taken
as the maximum of all the vent temperatures in the enclosure, i.e.
Tr = Max(Tvent)
Equation 4-17.
Centripetal and Coriolis forces
For flows in a rotating frame of reference (RFR), additional forces are introduced in the momentum
equations, Eq. (2-2), due to the rotation. These are the centripetal and the Coriolis forces. In vector
form, the source term in the momentum equations is expressed as
Equation 4-18.
where
vector,
If
is the rotational velocity vector (rad/sec),
, and
,
is the velocity
is the location vector with respect to the axis of rotation,
, 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 the
RFR as the origin of the axis and to define point B as the end point. One can define:
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Source terms
Rwx=xB-xo; Rwy=yB-yo; Rwz=zB-zo
Equation 4-20.
then
Equation 4-21.
Note that the direction of the rotation depends on the position of point B with respect to the origin
of the RFR.
It is important to note that the velocity on a RFR is relative to the rotating frame. If the model has a
fixed 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
by the Darcy-Forchheimer law [26].
of the momentum Equation 2-2 as described
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.
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Chapter
Source
Chapter
4: 4: Source
termsterms
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 the
global Cartesian system.
The components of the velocity along the principal axes are expressed in the global Cartesian
velocity components as follows:
Equation 4-26.
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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 Cartesian
coordinate system:
Equation 4-27.
This is the source term for the momentum Equation 2-2 in the case of flow through porous materials
with orthotropic resistance. The resistance force is nonlinear and is re-computed at each iteration.
Heat sources
Simcenter Flow permits transient analysis with a constant or time-dependent heat source on the
fluid side. In such cases, the source term, Sh in Eqs. (2-7 and (2-8) ), corresponds to that heat
generation per unit volume.
The heat source is imposed on selected fluid elements through the use of a generic entity. This
feature was developed to simulate arc faults but can be used for any flow situation where a heat
source 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 ambient
fluid 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.
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Chapter
Source
Chapter
4: 4: Source
termsterms
If the water vapor density is greater than the density at saturation (i.e. 100% relative humidity), the
water 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 diffusivity
of the water vapor).
Evaporation corresponds to a positive flux of water vapor to the fluid, while condensation corresponds
to a negative flux.
This mass flux is a source term in both the mass conservation equation Eq. (2-1) and the scalar
conservation 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 condensation
also affects the overall enthalpy of the fluid mixture. The energy flux to the fluid mixture resulting from
evaporation 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)
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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 on
whether 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 field
boundary. 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 turbulence
model, 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 stationary
wall, the fluid velocity at the wall is zero, while for translating and rotating walls, it is non-zero. The
calculation 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 a
laminar flow has to be relatively well refined in order to get a good approximation of the velocity
gradient at the wall. For turbulent flows, it would be impractical to fully resolve the details of the
boundary layer or near-wall region. For this reason, a semi-analytical approach is used whereby
the 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 distance
from the wall.
The derivation of the near-wall relations for turbulent flow depends on several assumptions. These
are presented in section Assumptions.
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Chapter
Boundary
conditions
Chapter
5: 5: Boundary
conditions
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. From
Eqs. (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 for
the fixed viscosity model is estimated in the same fashion as for the rest of the fluid domain, through
Equation (3-5). For the mixing length model, Equation (2.28) is used with the appropriate mixing
length l from Equations (3-7) and (3-8) and the strain rate from Equation (3-12). In the case of the
k-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, the
near wall transport equation for ε is replaced by (see section Assumptions for details)
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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 and
approaches a linear profile, so that:
Equation 5-11.
is computed as for laminar flow. Typically this departure starts at around y+= 30 and is complete
around y+= 5. Some sort of smooth transitional or “buffer” region exists in between. It is necessary to
model 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 inappropriate
and will reduce accuracy. Using nodes to resolve the near-wall region requires
modifications to other aspects of the turbulence model, e.g. including low-Reynolds
number model terms, and requires many nodes. The considerations given here are only to
ensure 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 between
the two limiting functions. In what follows, a general form is chosen that can also be used for
thermal wall function.
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In addition to the linear relation in the viscous sub-layer, the wall function needs to be modified for
roughness effects. The log-law region is the region where the average velocity of a turbulent flow at a
certain 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 the
thermal 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.
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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. For
reference, 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)
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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 a
smooth wall (i.e. k+ = 0), it blends equal parts log-law and linear-law at a y+ of 8. The
z+ variable is used (z+ = y+ for smooth walls) so that the switch-over point is correct
for rough walls. Physically, for very rough walls, it is observed that the log-law region
extends to somewhat closer than y+= k+ and then quickly the velocity drops to zero
before 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 to
zero 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 allowed
across the wall), or convecting. In the former case, a heat flux of zero is imposed at the wall, while in
the 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 fluid
temperatures through
Equation 5-21.
where k is the thermal conductivity of the fluid and yf is the distance from the wall at which Tf is
evaluated.
For turbulent flows, the temperature variation in the thermal boundary layer follows the general wall
function 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 the
near-wall velocity scale u*:
Equation 5-22.
The general function Eq. (5-14) describes the universal boundary layer profile for temperature with
the following definitions of its arguments:
Equation 5-23.
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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 functions
proposed 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 are
captured implicitly by the wall when y+ is defined through u*. Roughness τω and near wall k will
increase 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 the
energy equation in a manner similar to the approach used for momentum. Equation (5-22) is
re-arranged to,
Equation 5-28.
Given a near wall fluid temperature, Tf, Eq. (5-26) can be used to compute qw which is substituted
directly into the finite volume energy equation at the wall.
For implementation clarity and consistency with the momentum treatment, the following manipulations
are made.
First, using the definitions of y+, Eq. (5-26) becomes
Equation 5-29.
which can also be written as
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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 the
energy equation is not temperature but enthalpy, h. In general the relationship between these two
is arbitrary. However, Eq. (5-29) can be modified so that an approximate dependency of qw on h
is retained, while not changing the converged answer for qw. So, denoting the “new” solution with
superscript “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 heat
transfer coefficient. For flows with significant viscous heating however (such as high speed flows), the
heat flux is not dependent on the difference between Tw and Tf but rather between Tw and a so-called
adiabatic wall temperature. This adiabatic wall temperature depends on the local flow conditions and
can be derived from the thermal wall function. Denoting LS as low speed and HS as high speed, the
general thermal wall function can be written as:
Equation 5-33.
where
Equation 5-34.
Equation 5-35.
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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 contribution
from 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 (for
low speed flow).
Combining equations (5-32), (5-35) and (5-37), we get:
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Equation 5-40.
which allows to express the wall heat flux as:
Equation 5-41.
of equation (5-41) represents the standard low speed flow wall heat flux
The term
while the term u*T*HS/T+HS is the contribution to the wall heat flux due to viscous heating. qw is
positive 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 adiabatic
wall 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, the
use of Equations (5-42) and (5-43) allows to keep the standard formulation for the heat transfer
coefficient given by:
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Equation 5-46.
The effects of viscous heating are then accounted for by correcting the fluid temperature so that
T*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 the
wall 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 h
correlation 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 along
a 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 the
implementation of a natural convection velocity wall function. Implementing such a wall function
however, has implications in the treatment of the production and the dissipation of the turbulence
kinetic energy in the k-ε turbulence model. Only the temperature wall function for natural convection
is implemented.
Description of the temperature wall function for natural convection
Using temperature measurements for turbulent natural convection along a vertical plate in air, Yuan et
al. [23] were able to correlate the temperature profiles for a wide range of Re numbers using new
temperature 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.
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•
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 wall
function more general, the effect of Pr as well as the influence of the surface angle with respect
the 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 Tsuji
and 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 that
for any Pr number and for any surface angle, the non-dimensional temperature profiles in a natural
convection boundary layer are similar, it is possible to express equations (5-49) and (5-50) in a
more general fashion, using
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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 is
dependant 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)]max
Equation 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
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There is a certain degree of uncertainty in the determination of the Cu and Cv coefficients. Incropera
et al. [25] recommends a value of 0.15 for Cu while Raithby et al. [24] proposed a correlation which
accounts for an influence of the Pr number. In Simcenter Flow, the correlation of Raithby is adopted
with a small modification to make it compatible with the value of 0.15 for air. The Cu correlation
adopted is:
Equation 5-58.
As to the Cv coefficient, there are significant discrepancies between the correlation proposed by
Raithby et al. [24] and what is found in other heat transfer handbooks. The Raithby relationship for
Cv is:
Equation 5-59.
For Simcenter Flow, a new correlation for Cv as a function of Pr is derived so that the Nusselt
relationship Nu = CvRa1/3 matches as close as possible the more complex relationship recommended
by 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 Cv
coefficient is adjusted for each Pr number in order to minimize the error with eq. (5-60).
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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
Figure 5-2. Nu versus Ra for a vertical flat
plate
Although only three Pr numbers are illustrated here, a total of eight values were used ranging from
Pr = 0.1 to Pr = 2000 with, on average, a difference of less than 4% between eq. (5-60) and the
simple 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).
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Cv estimated from eq. (5-60)
eq. (5-61)
eq. (5-59)
Figure 5-3. Variation of Cv with Pr
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:
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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 some
manipulations,
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 the
surface orientation angle, the heat transfer coefficient is computed from equation (2.133) and the
natural convection thermal wall function.
Special considerations when calculating h
There are uncertainties regarding the validity of the correlations found in the literature when the
orientation of the solid surface is such that phi > 150 degrees. In addition, as the angle phi
approaches 180 degrees (heated surface facing downward or cooled surface facing upward), laminar
flow 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 be
done 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 temperature
Tn using:
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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 temperature
TN using:
Equation 5-70.
which gives:
Equation 5-71.
It can be shown, after several manipulations of equation (5-71) that the turbulent heat flux
corresponds 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 Simcenter
Flow indicate that the turbulent heat flux tends naturally towards the laminar heat flux for cases where
phi > 150 (heated surface facing downward or cooled surface facing upward).
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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 flow
rate, mass flow rate or velocity), or calculate the fan's flow rate from a fan curve definition. Depending
on 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 and
the 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 a
head loss. The software calculates this variable using Eq. (5-93).
A typical fan curve is illustrated in Figure 5-4.
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Fan's static pressure
Fan's volume flow rate
Fan curve
System curve
Fan's operating point
Figure 5-4.
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 rate
at 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 linear
system coefficients from the fan curve derivative based on the fan's pressure
rise at previous iteration (n). Therefore, both fan's pressure rise and fan's
volume flow rate are the results of the linear system solution.
Explicit approach
In the explicit approach, the solver determines the volume flow rate at current
iteration (n+1) directly from the fan curve itself using the fan's pressure rise
at previous iteration (n), and imposes the volume flow rate as an explicit
boundary condition to form the linear system. The software then obtains the
pressure 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 through
explicit iterations. This could lead to strong fluctuations in the solution convergence based on
the characteristics of the fan curve. You have to control these fluctuations using the relaxation
techniques. The implicit approach suppresses these fluctuations by making the pressure rise and
volume 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 rate
for 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.prm
file 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 and
volume flow rate using the Taylor series expansion.
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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,
the solution at iteration level, n. The updated volume flow rate,
current iteration level, n+1.
, are evaluated from
, is evaluated implicitly at the
In the implicit approach, the solver takes the fan curve data local to elements over the two faces on
opposite 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
from the derivative of the fan curve at the specified operating
element and obtains
point. In this approach, the solver inserts the local velocity into the solution matrix at a particular
point based on the pressure difference across a face element. This gives a relationship between
the 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 by
Eq. (5-75).
Equation 5-75.
The superscripts
n
and
n+1
refer to the previous and current iteration levels, respectively. The
Δpn
pressure rise,
and
are evaluated at iteration level, n. The updated volume flow rate,
, is
evaluated at the current iteration level, n+1. This volume flow rate is then imposed as the boundary
condition for the next iteration and a new value of pressure rise is calculated. However, when using
the explicit approach, the pressure rise and the volume flow rate at a given iteration level are out of
sync until the solution starts to reach some level of convergence, which can lead to some fluctuations
in the solution convergence.
The solver then uses the calculated volume flow rate through the fan to obtain the normal velocity
vectors on both faces of the fan: s1 and s2. The normal velocity vectors,
Dirichlet boundary conditions on the opposing sides.
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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 opposite
sides of the fan.
Equation 5-79.
The fan's pressure rise is then modified based on the fan curve type which is discussed in detail
in 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 actually
the 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.
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The software calculates
using Eq. (5-79). Having the fan's modified pressure rise, ΔP', the solver
obtains the corresponding fan's volume flow rate using the fan curve.
Equation 5-82.
Pressure rise calculation for outlet fan curve
The fan's pressure rise is defined by Eq. (5-76).
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 pressure
values on the interior nodes. Therefore, to form the intake total pressure, the solver adds the intake
dynamic 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 fan's pressure rise is defined by Eq. (5-76).
The pressure values at the intake of an internal fan are obtained from the static pressure values on
the interior nodes. Therefore, to form the intake total pressure, the solver adds the intake dynamic
pressure 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.
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Pressure rise calculation for recirculation fan curve
The fan's pressure rise is defined by Eq. (5-76).
The pressure values at the intake of a recirculation fan are obtained from the static pressure values
on the interior nodes. Therefore, to form the intake total pressure, the solver adds the intake dynamic
pressure 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 flow
rate, the pressure rise produced by the fan is reduced by the head loss. In the fan curve, the pressure
rise 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 head
loss coefficient, ν2, that is defined in Eq. (5-93).
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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.
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Flow angle on fan
The flow through an inlet fan or an internal fan can be specified to be at an angle θ from the
fan normal. In such cases, the velocity component normal to the fan, Vn, is the velocity obtained
from the fan curve, velocity, mass flow, volume flow or pressure rise specification. The velocity
component parallel to the fan plane, Vt, is then calculated so that the total velocity vector is in the
specified flow direction, i.e.:
Vn = Vspec
Equation 5-96.
Vt = Vntan θ
Equation 5-97.
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Boundary conditions
Swirl on fan
Swirl can also be imposed on inlet and internal fans by specifying an axis of rotation and an angle
θ from the fan normal. At each computation point on the fan, the velocity is calculated such that it
has a component normal to the fan plane and a component tangential to that plane in the direction
of rotation.
The velocity component normal to the fan plane, Vn, is obtained from the mass flow rate, velocity, fan
curve or pressure rise specification, as usual, whereas the tangential component, Vt, is calculated from
Vt = Vntan θ
Equation 5-98.
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Boundary
conditions
Chapter
5: 5: Boundary
conditions
Energy equation in fans
Specified temperature
The fluid temperature can be specified as a boundary value on inlet fans only. The temperature
imposed 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 to
the return side of the fan, i.e.
∆T = Treturn — Textract
Equation 5-99.
This is equivalent to specifying the value of the return temperature
Treturn = Textract + ∆Tspec
Equation 5-100.
Heat generation
Heat generation can be specified on internal fans as well as on recirculation fans. The amount of
heat 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 the
fluid and ΔT = Texhaust-Tintake for internal fans, ΔT = Treturn-Textract for recirculation fans. The heat
generation condition is in fact applied as a temperature specification at the exhaust side or return
side of the fan.
Turbulence quantities on fans
The solver applies turbulence quantities only at inlet fans and on the return part of the recirculation
fans. Depending on the turbulence model and the specified turbulence quantities, the following
equations are used to compute the turbulence kinetic energy, k, and the dissipation rate, ε, or the
specific dissipation rate, ω:
Equation 5-102.
Equation 5-103.
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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.
•
ρ is the fluid density.
•
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 transport
equation for water vapor in air is in terms of the mass ratio. The conversions from relative humidity to
mass ratio and from specific humidity to mass ratio are described in Section 2-
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Boundary
conditions
Chapter
5: 5: Boundary
conditions
General scalar equation
An inlet value of the mass ratio may be specified at inlet fans.
Openings
Openings, also called vents, are pressure and temperature specified boundaries. The condition
imposed depends on the direction of the flow, i.e. if the flow is an inflow or an outflow. The direction
of 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 separately
from its neighbors.
For the parallel solver, all elements of an opening are either on an inflow vent or on an outflow
vent at the same time.
Inflow vent
Pressure
At inflow vents, you specify the total pressure, Pspec = Ptotal, but the solver
imposes 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 screen
or a filter at that opening. In such cases, the static pressure is computed as
follows:
Equation 5-110.
Temperature
5-30
The vent temperature is applied at inflows only. The temperature is either the
specified value or ambient temperature.
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Boundary conditions
Turbulence
quantities
The solver applies turbulence quantities only at inflow vents. Depending on
the turbulence model and the specified turbulence quantities, the following
equations are used to compute the turbulence kinetic energy, k, and the
dissipation rate, ε, or the specific dissipation rate, ω:
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:
where:
•
U0 is the mean flow velocity at the boundary condition.
•
μ is the fluid dynamic viscosity.
•
ρ is the fluid density.
•
ΔP is the maximum pressure difference of the complete fluid domain.
•
A is the surface area of the vent.
Relative humidity or
specific humidity
The value of the relative humidity or of the specific humidity can be specified at
vents. This value is applied at inflows only.
Mass ratio for
general scalar
equation
An inlet value of the mass ratio may be specified at vents. This value is applied
at inflows only.
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Chapter
5: 5: Boundary
conditions
Angle from vent
An angle that is specified at a vent will affect inflows only. The velocity
component parallel to the vent, Vt, is calculated so that the total velocity vector
is 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 angle
between the flow direction and the vent normal.
Outflow vent
At outflow vents, you can only specify the pressure. The solver imposes the static pressure that
you specify:
Equation 5-112.
or, if the vent is to ambient
Equation 5-113.
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Boundary conditions
Convective outflow
The convective outflow boundary condition models fluid exiting the fluid domain without specifying
the pressure. It lets the flow field exit and enter the flow domain on each element of the boundary
as 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 dependent
variable 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 as
uniform, 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 the
dependent variables at the boundary face elements as a function of the values at the nodes of the
adjacent solid element, and imposes them implicitly.
For the mass equation on the convective outflow boundary, the outflow velocity field is computed
explicitly based upon the current nodal field. It is then scaled to conserve mass, accounting for
density variation with time.
Transient initialization
Because the temporal derivative is often dominant in the boundary value equation, a reasonable initial
condition is crucial; to this end, a small number of initializing steps are performed in which the outflow
boundary is treated as an exhaust opening, to obtain a reasonable and conservative starting point for
the solution of the evolution of the boundary field.
Steady state assumption
For better stability, in steady state runs, the advected values are computed from a
zero-normal-derivative condition, i.e. with temporal derivative explicitly neglected. This assumption
is true in the converged steady state solution. It removes the need for the initializing iterations and
improves convergence and robustness. Because the conservation of mass is explicitly enforced
over the entire domain, the pressure field at the outflow boundary can evolve naturally, so that the
upstream flow features are not strongly influenced by nearness of the boundary.
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Chapter
Boundary
conditions
Chapter
5: 5: Boundary
conditions
Supersonic inlet
In Simcenter Flow you can use a supersonic inlet boundary condition, where you specify the Mach
number, 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 specific
conditions.
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 treated
as compressible.
For low speed flows (Mach < 0.3) the energy equation is simplified. The pressure work and dissipation
terms 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:
•
•
5-34
is the velocity component normal to the screen.
f is the head loss coefficient.
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Boundary conditions
The direction of the velocity components on either side of the screen are identical, unless a flow angle
is specified at the screen. This flow angle and the pressure drop are accounted for in the mass
and 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 drop
to 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 choose
one 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.
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Chapter
5: 5: 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
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Boundary conditions
Rounded Edge
f'=0.03+0.47x10—7.7r/dh
The following restrictions apply:
•
Grid thickness is the same as orifice radius.
•
Red > 103.
Figure 5-7. Rounded edge orifice
Beveled Edge
f'=0.13+0.34x10f”
f''=-(3.4t/dh+88.4((t/dh)2.3)
The following restrictions apply:
•
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:
•
A0 is the area of a single orifice.
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Chapter
5: 5: 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:
•
A0 is the area of a single orifice.
•
AFs is the area of the free stream.
•
FAR = ΣA0/AFs is the free area ratio.
The following assumption applies: Red > 500.
Symmetry boundaries
A problem is symmetric about a plane when the flow on one side of the plane is a mirror image of
the flow on the opposite side of the plane. In such case, the use of a symmetry plane condition
enables efficient use of computer resources by allowing the numerical solution to be obtained on a
fraction of the original domain.
The application of a symmetry plane condition to a planar surface of the grid means that the solution
on 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 are
parallel to the plane of symmetry.
Periodic boundaries
Periodic boundary conditions are used to simulate a flow leaving through a boundary A and entering
through 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 A
and B were adjacent. Any pair of periodic boundaries must have similar shape and size, and can
either be parallel to one another, or not. In the later case, one boundary must be the copy of the other
periodic boundary, rotated an angle phi with respect to a specified axis of rotation.
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Boundary conditions
Non-parallel periodic boundaries can be used, for example, to simulate flow through a pipe without
having to model the entire 360 pipe cross-section: the user can model an angular section of the pipe
cross-section and impose periodic boundary conditions on the two artificial boundaries.
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Chapter 6: Particle tracking
Particle tracking model
Simcenter 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 wake
effects included in the model.
Note
Brownian motion is the apparent random movement of particles suspended in a fluid due
to turbulence.
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Chapter
Particle
tracking
Chapter
6: 6: Particle
tracking
Turbulent particle motion
The turbulent particle interactions are modeled following a Brownian dynamic approach and using
the 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, where
D 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 the
turbulence models into the particle difussivity as in Eq. (6-3).
D=(kbT)/(6πμrp) + (μt/ρ)
Equation 6-4. Particle tracking difussivity with turbulent effects
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Chapter 7: Discretization
Control-volume method
The governing partial differential equations described in 2- are discretized using a finite-element
based control-volume method, [5] with this method, the governing equations are integrated over a
control volume and over a time step. As an example taking the equation expressing the conservation
of 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 the
outer surface area of the control volume. The volume and surface integrations are approximated
numerically over a discrete finite volume defined on a computational grid or mesh. For instance, the
transport, 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 integration
point of this sub-surface, ΔA is the sub-surface area, ip is the discrete value of Φ at this sub-surface
and the sum is over all surfaces of the finite volume.
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Discretization
Chapter
7: 7: Discretization
Element
Node
Finite Volume
Integration point
Element sector
Figure 7-1.
In the element-based finite volume method, the finite volume is defined from elements, [5], as
illustrated in 7-1. All the dependent variables, including pressure and the velocity components, are
stored 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 finite
volume results from two surrounding quadrilateral elements and three triangular elements. These
sub-surfaces are called integration point surfaces, and the integrand to be evaluated is computed at
their 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 the
finite volume sectors, are indicated. The discretization scheme thus entails representing discrete
approximations to the volume integrals over the element sectors and discrete approximations to the
surface integrals on the element integration point surfaces. Robust, efficient and accurate schemes
to achieve this have been implemented. They are detailed for each governing equation in section
Details of the discretization for the mass conservation equation. The overall equation assembly
proceeds by visiting each element in turn, making the discrete approximations to the terms in the
integrals, so that when all elements have been visited, every node has a completed finite volume
equation for each conservation law.
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Discretization
Details of the discretization for the mass conservation equation
Upon 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 the
integration point velocities, [5]. The resulting form of the mass conservation equation contain a
pressure redistribution term, which resolves the pressure-velocity coupling problems that is typical
of colocated methods, [6].
Details of the discretization for the other conservation equations
The discretized form of the momentum conservation equation for velocity Uj , Eq. (2-2) can be
written 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.
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Discretization
Chapter
7: 7: 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 source
terms all have the same form from one equation to the other, and all are treated similarly in Simcenter
Flow. The treatment of each of those terms and of the pressure term of the momentum equations is
described 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 further
treatment.
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 the
boundaries of a given control volume. The quantity Φip has to be evaluated in terms of nodal Φ
values. Various schemes can be used to evaluate this quantity. These schemes are known as first
order or higher order schemes, based on the following simplified analysis:
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Discretization
Consider the 1D representation of a control volume and its immediate neighbors illustrated in figure
7-1. 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 Taylor
series only, then the truncation error is first order and the advection scheme is first order. Similarly, if
Φi+1/2 is approximated by the first two terms of the series, the truncation error is second order, and
so is the advection scheme.
The default advection scheme in Simcenter Flow is first order. Four higher order schemes with flux
limiters are also available.
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Discretization
Chapter
7: 7: Discretization
First order scheme
The first order scheme, upwind differencing scheme (UDS) or differencing scheme, Φip is
approximated 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, the
solution is very stable and converges quickly. UDS can, however, give good results only if the flow is
aligned 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 error
introduced 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 order
schemes can obtained from Eq. (7-13) depending on how the gradients in the Taylor series are
evaluated.
Higher order schemes help reduce considerably the problem of false diusion, but they still have a
truncation error. In the case of second order schemes, this truncation error is dispersive, while for
third order schemes, it is dissipative. In both cases, this may lead to unphysical oscillations and
numerical 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 expensive
in terms of calculation time.
In order to eliminate the oscillations that are inherent to most higher order schemes, it is possible to
impose 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 limiter
constraints, are conditions that act to limit the face value, Φi+1/2, to lie between specified values. The
concept of flux limiters is similar in many ways to the TVD schemes (Total Variation Diminishing
schemes) that are widely used in gas dynamics [14].
Higher order schemes
Higher order schemes are available in Simcenter Flow. The recommended scheme is SOU (Second
Order Upwind); this is also the default high order scheme. The other schemes available are
UDS+NAC (Upstream scheme + Numerical Advection Correction), UDS+CDS (Upstream scheme +
Central Dierence Scheme correction) and QUICK (Quadratic Interpolation scheme). The details of
those 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. In
some cases such as turbulent flows being solved using the k-ε model, a solution is impossible to
get without flux limiters. Details of the ux limiters used in Simcenter Flow are given in the following
sub-section.
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Discretization
Flux limiters
In general, a limited higher order estimate for the value Φip of the dependent variable Φ at an
integration point whose upwind and downwind neighbors are nodes C and D respectively, may be
expressed 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 UDS
scheme and the higher order scheme, ideally selected to be the largest possible value which
effectively prevents the development of undesirable artifacts, sush as unbounded values or spurious
oscillations, within the solution.
The limiters used for the SOU, QUICK, and CDS schemes in Simcenter Flow are based on the
Convective Boundedness Criterion (CBC). The details of the implementation of this scheme are given
in [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 approximation
based upon the cell-centered gradient ÑΦC at the upwind node is employed, according to the
following equation:
Φip,NAC = ΦC + (xip - xC)·ÑΦC
Equation 7-18.
The limiter is determined from the differentiable replaceable of the familiar Barth and Jespersen
limiter, 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.
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Chapter
7: 7: Discretization
where the maximization in the first equation of the Equation 19 is carried out over all intergration
surfaces bounding the control volume about node C, δ is the average mesh spacing, and Φext is
an 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 the
minimum 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 the
magnitude 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 parameters
defining the transport problem of interest, with the intent of effectively damping the growth of spurious
extrema while not unduly hindering convergence.
Diffusion terms
The 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 derivatives
within an element. Given an (s, t, u) coordinate within the element, the value of the field variable, for
example , 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 the
element nodes, an appropriate polynomial in (s, t, u) and that the shape function corresponding to a
particular 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 the
derivatives of the shape functions as follows:
Equation 7-22.
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Discretization
Source terms
The discretized source terms are simply evaluated by multiplying Suj, Sh and S by the volume of the
control volume, as expressed in Eqs. (7-7) to (7-12).
Pressure term
The pressure term in Eq. (3.6) ,
at the integration point ip is determined from
, is evaluated using shape functions. The pressure
Equation 7-23.
where Pnode are the nodal pressure values.
Rotating frames of reference
The frozen disk methodology used for rotating frames of reference is useful for models where the
flow distribution around the interface between the fixed and the rotating frames is non-uniform. The
interface between the frames is called a RFR Interface and takes into account the frame change with
a 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 for
the 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 is
chosen 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 the
components of the location vector with respect to the origin of the RFR.
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Chapter
Discretization
Chapter
7: 7: Discretization
Node 1
Rotating Frame
Node 2
Rotating Frame
Figure 7-3.
The equations for nodes 1 and 2 are first assembled considering all the fluxes except the ones
through the RFR interface. Labelling the fluxes through the screens as Q1 and Q2, the control volume
equations 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 B2
represent 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 the
mass 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.
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Discretization
Mixing Plane
A mixing plane boundary condition interfaces two or more fluid volumes with different flow conditions
in 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 the
adjacent fluid volume as boundary conditions.
The interfacing faces are subdivided in a number of areas you specify. The areas are matched to its
closest corresponding area according to their geometry and the averaging method you choose.
Table 7-1. Averaging method examples
Along radius
Along vector
Upstream
Averaging direction
Downstream
Number of segments (Ex. 5)
For each area, the pressures and velocities are transformed to a 1D field using an averaging
technique 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 the
interfaces.
You define either a radial or axial averaging method.
•
When the defined axis is perpendicular to the mixing plane, the velocity and the pressure are
averaged along radial arcs at the interfaces.
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Chapter
Discretization
Chapter
7: 7: Discretization
•
7-12
When the defined axis is parallel to the mixing plane, the velocity and the pressure are averaged
along axial segments at the interfaces.
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Chapter 8: Solver numerical method
Solver numerical method
Once the discrete finite volume equations have been assembled, they must be solved. The
implementation assembles and solves the mass and momentum equations together, making it a
coupled, 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. As
this is not a direct solver, it is used within an iterative refinement loop, in which the exact solution is
approached 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 is
to 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 algorithm
will 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 balances
of conserved quantities over a finite volume. Suitable merging of the original finite volumes to create
larger ones than create the coarser grid equations. This is done recursively, so that the hierarchy is
created. The coarse grid equations thus impose conservation over a larger volume and by so doing
reduce the error components at longer wavelengths. This is illustrated in Figure 8-1, where the nodes
and finite volumes (not elements) are shown.
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8-1
Chapter
Solver
numerical
method
Chapter
8: 8: Solver
numerical
method
Figure 8-1.
In this hierarchy, current fine grid residuals are passed down to the next coarser grid, denoted by the
down arrows, with additional corrections to the relatively finer grid solution being obtained from the
coarser grid equations, denoted by the up arrows.
The ILU iterative solver improved by this Multigrid accelerator results in a complete linear solver that
demonstrates no degradation of performance in the presence of large aspect ratios, and only a
linear increase in cost with problem size. The current method employs a coupled algebraic Multigrid
accelerator to prevent this behavior, [10].
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Chapter 9: Temporal discretization
Temporal discretization
For most of the transport equations, you can select either a fully-implicit (backward Euler) or
semi-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.
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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 and
Fluid 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 with
Trailing 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. 10th
GAMM 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 related
exponential schemes for convective modeling - There are much better alternatives", International
Journal 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 New
Boundedness-Preserving Transport Algorithm", International Journal for Numerical Methods
in 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
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Chapter
References
Chapter
10:10:References
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 Transfer
90, 1-8 (1968).
22. T. Tsuji and Y. Nagano, Velocity and temperature measurements in a natural convection boundary
layer 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 convection
along 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, John
Wiley, Inc. (1996)
26. S. Whittaker, Volume Averaging of Transport Equations. In J. Prieur du Plessis (ed.). Fluid
Transport in Porous Media. Chap. 1. Computational Mechanics Publications: Southampton,
UK (1997).
27. V. Venkatakrishnan, Convergence to Steady State Solutions of the Euler Equations on
Unstructured 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.
10-2
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Index
C
Condensation and Evaporation . . . . . . . . 4-7
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Index-1
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