6819874 Turbulent Flows

8/8/2019 6819874 Turbulent Flows

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MODELING TURBULENT FLOWS Introduction

Fluent Inc., Dec-98 7-1

7. MODELING TURBULENT FLOWS

7.1 Introduction

FIDAP provides extensive capabilities for the simulation of turbulent flows. This chapter

summarizes these capabilities and the steps required to set up and run a turbulent simula-

tion using FIDAP. Details on the theoretical background of both the equations solved and

the numerical techniques employed can be found in the FIDAP Theory Manual.

The purpose of this chapter is to help you make optimum use of the many FIDAP com-

mands and options available for obtaining convergent and realistic solutions to turbulent

flow problems. Some of the parameters that affect physical and numerical aspects of typi-

cal turbulent flow simulations are as follows:

Boundary conditions

Mesh density and spatial distribution

Solution procedure

FIDAP requires only a few main commands and keywords for simulating turbulent flows.

The following table summarizes these commands.

Command Keyword Description

PROBLEM TURBULENT Indicates that the ensemble- or time-

averaged flow equations are to be solved.

Because these mean equations contain the

unknown turbulent fluxes, it is necessary

to select an appropriate turbulence model

by means of the VISCOSITY command

(see below).

VISCOSITY MIXING Specifies the use of a zero-equation type

turbulence model with the mixing length

computed by means of a user-subroutine,USRMXL.

MIXLENGTH= v Specifies the use of a zero-equation type

turbulence model with automatic compu-

tation of the mixing length.

TWO-EQUATION Specifies the use of a two-equation type

turbulence model.


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Introduction MODELING TURBULENT FLOWS

7-2 Fluent Inc., Dec-98

Command Keyword Description

TURBOPTIONS STANDARD

EXTENDED

ANISOTROPIC

RNG

K-OMEGA

Specifies a particular two-equation type

turbulence model.

EDDYVISCOSITY BOUSSINESQ

SPEZIALE

LAUNDER

Specifies the eddy-viscosity constitutive

relation to use in the turbulence model.

This chapter presents information related to two different types of turbulent flow models:

Zero-equation (Section 7.2)

Two-equation (Section 7.3)

The bulk of the chapter is devoted to the use of the two-equation models. The reasons for

this focus are as follows:

Two-equation models are more accurate and universal in their application and are

employed more frequently than are zero-equation models.

There are considerably more physical and numerical issues involved in performing

turbulent flow simulations with a two-equation model, therefore it is more difficult

for the inexperienced user to obtain optimal solutions to such simulations.

Invoking a two-equation model entails the solution of two additional transport equations

which can significantly increase the CPU requirements of the numerical solution. In addi-

tion, the introduction of the k and A (or M ) equations significantly increases the nonline-

arity and coupling of the overall flow equations and, in general, destabilizes the con-

vergence characteristics of the numerical solution. By contrast, turbulent flow simulations

using zero-equation type models (where the value of the eddy viscosity is either fixed or

prescribed algebraically) exhibit relatively stable convergence characteristics similar tothose observed in the simulation of laminar flow problems.


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MODELING TURBULENT FLOWS Introduction


To successfully use FIDAP to model turbulent flows, you must possess an adequate

knowledge of such flows and of turbulence models. Such knowledge will enable you to

choose the optimum turbulence model based on accuracy requirements and available CPU

resources. It will also allow you to correctly evaluate the solution to determine whether or

not the results are realistic. The task of verifying the solution is non-trivial and often

entails performing additional runs to check the sensitivity of the solution to boundary

conditions (such as the boundary conditions ofk and A at the inlet plane to the computa-

tional domain) as well as other physical and/or numerical parameters. Sometimes it isnecessary to develop a feel for the numerics and the physics of turbulent flow simula-

tionsstarting from simple flow problems and moving toward more difficult ones.


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Zero-Equation Flow Simulations MODELING TURBULENT FLOWS


7.2 Zero-Equation Flow Simulations

Zero-equation models employ only algebraic equations to describe the relationship

between the turbulent viscosity,m t, and the tangible flow quantities. Although there are

many different forms of zero-equation models, FIDAP employs a model known as the

mixing-length model. In the mixing-length model, m t is computed using an equation of

the form

m rt m i j j i i jl u u u= +2

, , ,c h . (7.2.1)

(For a complete description of the significance and use of equation (7.2.1), see Chapter 10

of the FIDAP Theory Manual.)

Equation (7.2.1) involves a single unknown parameter known as the mixing length, lm ,

which can be thought of as the mean free path for the collision or mixing of globules of

turbulent fluid. The distribution of lm over the flow field must be prescribed with the aid

of empirical information.

The mixing-length model works well for relatively simple flows such as thin shear-layer

flows, wall boundary-layer flows, jets and wake flows, because lm can be specified by

simple empirical formulas in such situations. However, the model does not account for thetransport and history effects of turbulence. In particular, the model is not suitable when

processes of convective or diffusive transport of turbulence are importantsuch as in

rapidly developing flows, heat transfer across planes with zero velocity gradient, and

recirculating flows. More generally, the model is often difficult to apply in complex flows

because of the difficulties in specifying lm .

From a mathematical standpoint, a system of flow equations resulting from a zero-

equation type turbulent flow simulation is virtually identical to a system of laminar flow

equations with a constant or variable molecular viscosity. (Examples of laminar variable-

viscosity flows are flows that involve a temperature-dependent viscosity or non-Newto-

nian flows with shear-rate dependent viscosities.) Thus, as far as numerical aspects and

convergence characteristics are concerned, zero-equation type turbulent flow simulations

perform similarly to laminar flow simulations. In particular, if a well-defined set ofboundary conditions is employed, the numerical solution to a zero-equation type flow

model generally converges at a fast rate and produces a finely converged result in a rela-

tively small number of iterations. Additionally, the convergence characteristics of the

numerical solution are relatively insensitive to the following factors:

Changes in the density and spatial distribution of the mesh

Changes in the shape of the flow domain

The starting guess solution (that is, the radius of convergence is relatively large)


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MODELING TURBULENT FLOWS Zero-Equation Flow Simulations


The accuracy of the simulation, however, depends strongly on the adequacy of the zero-

equation turbulence model.

7.2.1 Mixing-Length Computation Methods

FIDAP provides two methods of specifying a zero-equation turbulence model:

Automatic mixing-length computation

User-subroutine mixing-length computation

The following sections describe each of these methods.

Automatic Mixing-Length Computation

When you specify the automatic mixing-length computation, FIDAP calculates the mixing

length at every point in the flow according to the following equation:

l l lm n c== min , .k 0 09 (7.2.2)

where ln is the distance from the nearest wall, and lc is a characteristic length scale of the

flow. (For example, for an internal flow, lc is the half-width of the channel.) For any

given flow problem, FIDAP computes one global value of lc that is equal to the maximum

value of ln in the entire mesh. Although this global value is appropriate for problems

with simple geometries, it may not be representative for all flow regions in more compli-

cated geometries. For this reason, FIDAP provides the flexibility of overriding the value

oflc in selected flow regions.

To activate the automatic mixing-length computation, you must specify the keyword

MIXLENGTH = v on the VISCOSITY command, where the value v is the characteristic

length scale, lc.

If you specify a MIXLENGTH value of zero, or do not specify a value, FIDAP

employs the computed global value of lc .

If you specify a non-zero value for v , FIDAP overrides the computed global valueoflc.

If you define VISCOSITY model sets for two or more fluids, the MIXLENGTH keyword

allows you to explicitly specify the characteristic length scale lc for the different FLUID

entities in various regions in a geometrically complex flow.


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Zero-Equation Flow Simulations MODELING TURBULENT FLOWS


User-Subroutine Mixing-Length Computation

In lieu of the automatic computation of mixing length, FIDAP allows you to explicitly

specify the mixing-length computation by means of the user subroutine, USRMXL. To

employ a user-subroutine mixing-length computation, you must specify the MIXING

keyword on the VISCOSITY command and provide the USRMXL subroutine. For a

complete description of the USRMXL subroutine and its use, see Chapter 9 of the

FIPREP Users Manual. (NOTE: A large number of zero-equation type turbulencemodels are available in the literature.)

7.2.2 Specification Procedures

The steps required to specify a zero-equation turbulence model are as follows:

Step Description

1 Specify a TURBULENT analysis on the PROBLEM command.

2 For an automatic mixing-length computation, specify the MIXLENGTH

keyword on the VISCOSITY command.

For a user-subroutine mixing-length computation, specify the MIXING

keyword on the VISCOSITY command and provide a subroutine

(USRMXL) to compute the mixing length.

3 (MIXLENGTH option only)

Specify WALL boundary entities on any portions of the boundary which are

not inflow, outflow or symmetry boundaries. (NOTE: FIDAP automatically

assumes that the boundary of any SOLID or POROUS entity is a wall for

the purposes of the mixing-length computations, therefore WALL entities

are not required on these boundaries.)


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MODELING TURBULENT FLOWS Two-Equation Flow Simulations


7.3 Two-Equation Flow Simulations

Two-equation turbulence models employ two extra partial differential equations to

describe the relationship between the turbulent viscosity,m t, and the tangible flow quan-

tities. FIDAP provides five two-equation turbulence models and three eddy-viscosity

constitutive relations (see the following table).

Turbulence Models Eddy-Viscosity Constitutive Relations

Standard k- e (default)

Extended k- e

RNG k- e

Anisotropic k- e

Wilcox Low-Re k- w

Boussinesq (Default)

Speziale

Launder

For most cases that involve high-Re flows, the model defaults (standard k- e model and

Boussinesq constitutive relation) produce satisfactory results. For low-Re flows, the rec-

ommended practice includes the use of the k- w model and Boussinesq relation. For a

complete description of the models and relations listed above and their use in turbulence

modeling, see Chapter 10 of the FIDAP Theory Manual.

Employing a two-equation turbulence model significantly increases the required CPU

resources for the solution of a given flow problem compared to the corresponding simula-

tion using a zero-equation type model. Moreover, because of the significantly stronger

nonlinear and intercoupled nature of the system of flow equations, it becomes somewhat

more difficult to obtain a fully converged solution. In general, the convergence character-

istics of two-equation simulations are less stable than are those of zero-equation simula-

tions and are somewhat sensitive to the numerical and physical parameters involved in the

numerical solution. In the following subsections, these various parameters are highlighted

and some guidelines are provided as to how to best manipulate them (by means of the

available FIDAP commands) to obtain a stable numerical solution.


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Two-Equation Flow Simulations MODELING TURBULENT FLOWS


7.3.1 Boundary Conditions

One important difference between two-equation simulations and zero-equation simula-

tions is that, for a two-equation simulation, appropriate boundary conditions must be pre-

scribed on the boundaries of the computational domain for the primary turbulence

variablesthat is, k and A (or M ). In general, five types of computational boundaries are

encountered in typical flow simulations (see Figure 7-1). They are:

Inlet

Outlet

Symmetry

Wall

Entrainment

Figure 7-1: Typical computational flow domain boundaries

The following subsections describe the boundary conditions that must be applied for the

primary turbulence variables at the boundaries listed above.

NOTE: To specify a boundary condition for the turbulent frequency ( M ) for the k- M

model, you must input the corresponding value for the dissipation, A ( == kM ) by means of

the DISSIPATION keyword on the BCNODE command. When you specify the use of the

k- M model, FIDAP automatically converts the input A value to its corresponding turbu-

lent frequency ( M A==

k).


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Inlet boundaries

Inlet boundaries are typically positioned upstream of the regions of interest and are

located in areas where the flow field is unperturbed by any nearby obstacles. Dirichlet

(that is, prescribed or essential) boundary conditions for k and A (or M ) must be em-

ployed on inlet boundaries.

Strictly speaking, the levels and the shapes of the profiles ofk and A (or M ) at the inlet

boundary are unique for every flow problem. Ideally, they should be obtained from

experimental measurements, but such experimental data is rarely available for typical

simulations. Moreover, a precise set of laws does not exist from which appropriate

profiles for k and A (or M ) can be derived for all possible flow scenarios.

The two most commonly occurring situations are as follows:

External/unconfined flows

Internal or partially confined flows

The following paragraphs describe the procedures required to obtain reasonable estimates

for the characteristic k and A scales for each of the flow situations listed above.

External/Unconfined Flows

Estimating k

To obtain a reasonable estimate of k for external or unconfined flows, you must first

specify a characteristic velocity scale of the mean flowfor example, the free stream

velocity, u

. Then you can obtain the characteristic value ofk from

k au=

2

(7.3.1)

where a is a turbulence intensity which assumes the following range of values

aOO~

( ~ . )( . )

-

0 0 00 10 1{

; free- shear flows (wakes, jets,etc.)

; shear - free flows

In some shear-free situationsas is often the case in wind tunnel flowsan alternative

definition of turbulence intensity, I, is used. In such cases, the turbulence intensity is

defined as

Iu

u=

21

2

e j(7.3.2)

and a value for k can readily be arrived at from the following expression,


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k Iu=

1 52

. b g . (7.3.3)

Estimating A

You can obtain the characteristic value of A in one of two ways, depending on whether a

characteristic eddy length scale d l can be identified at the inlet boundary (free-shear flow)

or not (shear-free flows).

If the eddy length scale can be identified, e can be obtained from the expression

e

d d

= -

k k

l

32

32

0 1~

.(7.3.4)

where d is the characteristic width of the shear layer at the inlet plane (see Figure 7-2).

If the eddy length scale cannot be identifiedthat is, a characteristic length scale of the

mean flow or turbulence fields is not availablea characteristic value of e can be

obtained from the expression

e r

m

m

m

=

=

ck

R

2

(7.3.5)

where R tm m m= is the ratio between the turbulent and laminar viscosities. In general,

Rm

varies considerably from one flow problem to another. However, for most problems a

reasonable value of Rm

can be arrived at after some trial and error. For example, for

free-shear flows with very low levels of turbulence R Om

~ 1a f . For most typical flows Rm

~

O(101)O(10

2).

Specifying k and e

The characteristic values of k and e obtained from the procedures listed above can be

specified as Dirichlet boundary conditions at the inlet plane by means of the KINETIC

and DISSIPATION keywords, respectively, on the BCNODE command. Moreover, if

equations (7.3.1) and (7.3.4) are used to arrive at characteristic inlet k ande

values, theycan be applied simultaneously together with the inlet velocity boundary condition using

the INTENSITY and LENGTH keywords of the BCNODE command. The value of the

INTENSITY keyword on the BCNODE command is the ratio of k to the square of the

characteristic velocity scale, u

, expressed as a percentage. Thus, with respect to equation

(7.3.1), the value of the INTENSITY keyword equals a 100. The value of the LENGTH

keyword is the size of the characteristic eddy length scale,d l , that appears in equation

(7.3.4).


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Figure 7-2: Schematic definition of @ , the shear layer width, for various types of flows


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Internal or Partially Confined Flows

Estimating k and A

In the case of internal (fully confined) or partially confined flows (for example, boundary

layer flows), the following expressions can be used to arrive at appropriate profiles for k

and A :

k c ld u

d ym= FHG IKJm

12

2

(7.3.6)

A

m

=

FHG

IKJ

-

c k ld u

d ym

2 2

1

(7.3.7)

where lm is a mixing-length expression appropriate to the current model, u is the stream-

wise velocity component at the inlet plane, and y is the normal coordinate axis to the

nearest wall (see Figure 7-3).

If the inlet plane is located in a region that is remote from flow obstructions, the stream-

wise velocity profile at the inlet plane can be closely approximated by a power-law profile

of the form

u

u

y

r

h

=

FHG

IKJ

d

1

, (7.3.8)

where the exponent h is Reynolds number dependent and is obtained from the following

table.

r d m

u r h

4 103

2.3 104

1.1 105

1.1

106

2 106

3.2 106

6.0

6.6

7.0

8.8

10

10


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uy

y

ur

u

y

ur

Figure 7-3: Flow configuration at inlet boundary for fully- and semi-confined flows

Specifying k, A , and u

The above profiles ofk, A and u can be imposed as Dirichlet boundary conditions on the

inlet boundary in FIDAP using the user-subroutine option of the BCNODE command.

For a description of the BCNODE command and related user subroutines, see Chapter 7

of the FIPREP Users Manual.


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Outlet and Symmetry Boundaries

The Neumann (that is, the zero-gradient or zero-flux) boundary condition is the most

appropriate for k and A on outlet and symmetry boundaries. As part of the finite element

discretization, FIDAP automatically assigns Neumann boundary conditions on portions of

the computational boundaries to those flow variables that have not been explicitly given

Dirichlet or gradient boundary conditions by means of the BCNODE and BCFLUX

commands, respectively. Thus, in the context of FIDAP, Neumann boundary conditionsare imposed on k and A on a given outlet or symmetry plane if no explicit boundary con-

ditions for k and A are applied at that location.

CAUTION: In designing the computational mesh, you must ensure that the outflow

boundary is placed at a downstream location that is sufficiently far from regions of the

flow where large perturbations occur in the flow field. For example, if flow over a back-

ward facing step is being simulated, the flow exit plane of the computational domain

should be placed sufficiently downstream of the point of reattachment to avoid any possi-

ble interaction between this boundary and the flow patterns in the recirculation zone.

Specifically, in the case of turbulent flow over a backward facing step, the stream-wise

extent of the recirculating bubble (that is, the reattachment length) is approximately seven

step heights. The exit plane must be placed at least one and a half recirculating lengths

(that is, about 11 step heights) downstream of the reattachment point.

Wall Boundaries

In turbulent flows that are bounded by walls, the near-wall modeling methodology must

be applied along those portions of the computational boundary that coincide with the solid

walls. This specialized methodology is completely transparent to the FIDAP user. (For a

complete description of the near-wall modeling methodology, see Chapter 10 of the

FIDAP Theory Manual.)

To apply the near-wall model, you must assign a WALL boundary entity along the portion

of the computational boundary that coincides with the solid boundary. When a WALL

entity is assigned, FIDAP searches all FLUID entities and identifies, as special near-wall

elements, those FLUID elements the sides of which coincide with the WALL boundary

elements. You must also apply the appropriate boundary conditions for those mean flow

equations (for example, momentum, temperature, and/or species) that are being solved as

part of the flow model.


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Momentum Boundary Conditions

To specify a problem that involves an isothermal flow with a stationary solid wall, you

must assign zero values for velocity components at the wall. If moving walls are present

and/or if there is transpiration at the wall, the appropriate non-zero velocity boundary

conditions must be imposed at the wall. (NOTE: If you prescribe non-zero velocity

boundary conditions along WALL boundaries, the velocity components on any node lying

on these boundaries should be entered in terms of the global coordinate system, ratherthan a local normal-tangential system such as is used for SLIP boundaries.)

Temperature and Species Boundary Conditions

If heat or mass transfer is present in the problem, an appropriate boundary condition for

the temperature or species equation is needed. The boundary condition can take the form

of a prescribed temperature (or species concentration) or a prescribed heat (or mass) flux.

In the case of a conjugate heat and/or mass transfer problem, where the boundary between

the fluid and solid regions is internal to the computational domain, no explicit boundary

conditions need to be applied at the fluid/solid interface for the energy or species equa-

tions, because the values of temperature and species concentration at this interface are

computed as part of the numerical solution.

NOTE: When WALL boundary entities are employed in a model, FIDAP uses a boundaryunit normal vector to compute the characteristic cross-flow widths of the special near-wall

elements. On external boundaries of the computational domain, the direction of the unit

normal vector is uniquely defined as pointing away from the computational domain. On

internal boundaries, however, the direction of this normal vector is not uniquely defined,

and you must provide a direction. For internal boundaries involving WALL entities, you

should orient the unit normal vector such that it is pointing away from the FLUID entity

and toward the SOLID entity. This orientation is achieved by means of the ATTACH

keyword when the WALL boundary entity is defined using the ENTITY command. The

value of the ATTACH keyword must be set to the name of the FLUID entity. For a

detailed description of the ATTACH keyword, see Chapter 6 of the FIPREP Users

Manual.

Porous Media Boundary Conditions

Internal interface boundary conditions for turbulent flows that involve porous media (with

or without heat/mass transfer) are similar to those described above. However, an fluid-

porous interfaces differ from fluid-solid interfaces in that they involve flow across the

interface. Thus, no velocity boundary condition should be explicitly applied at such an

interface, because the velocity at that location is computed as part of the numerical

solution. If significant shearing of the flow is anticipated at the fluid-porous interface, it is

recommended that WALL boundary entities be employed along the interface.


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Boundary Conditions on k and A

It is unnecessary to apply k and A boundary conditions on the wall portion of the com-

putational domain, because this is done automatically within FIDAP as part of the near-

wall model. The k and A boundary conditions that FIDAP applies are described in

Chapter 10 of the FIDAP Theory Manual.

Entrainment Boundaries

Entrainment boundaries typically occur in external or semi-confined flow configurations.

There are two common types of entrainment boundaries, each of which can be distin-

guished from the other by the size of the ratio of the entrainment velocity to the tangential

velocity. The two types are as follows:

Small ratio (see Figure 7-4)

Large ratio (see Figure 7-5)

Small-Ratio Entrainment Boundary

Figure 7-4 shows two examples of boundaries that are characterized as having a small

entrainment velocity component (that is, the velocity component normal to the boundaryplane) relative to the tangential velocity component. The figure shows that entrainment on

such boundaries can be both in andout of the flow domain.

Zero-traction boundary conditions are appropriate for the components of the momentum

equation. The appropriate boundary conditions for k and A on these boundaries are

Neumann (that is, zero-gradient) conditions. In some cases, harder Dirichlet boundary

conditions for k and A can be applied on these entrainment planes (the characteristic

values ofk and A employed at the inlet plane can be used for this purpose), but these are

not as appropriate as the Neumann boundary conditions.


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Figure 7-4: Entrainment boundaries on which entrainment velocity is small

compared to tangential velocity


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Figure 7-5: Entrainment boundary on which entrainment velocity is large

compared to tangential velocity

Large-Ratio Entrainment Boundary

Figure 7-5 shows an entrainment boundary that is characterized as having a large entrain-

ment velocity that points into the flow domain and is considerably larger than the tangen-

tial velocity component. The appropriate boundary conditions for the velocity components

are a zero tangential velocity component and a zero normal traction condition for the

normal velocity component. The appropriate boundary conditions for k and A are

Dirichlet conditions characterizing the state of turbulence in the free stream regions

outside the flow domain. Equations (7.3.1) to (7.3.5) can be used to arrive at the charac-

teristic free stream values ofk and A . If the free stream is totally turbulence free (that is,

if it is laminar), k and A can be set to zero on the entrainment boundary.

7.3.2 Initial Conditions for k and AA

Experience has shown that the convergence characteristics of most k- A runs are consid-

erably improved when non-zero initial guess fields are used for k and A . The recom-

mended approach is therefore to impose constant non-zero initial fields ofk and A via the

KINETIC and DISSIPATION keywords on the ICNODE command, irrespective of

whether a steady-state or transient simulation is being performed. The initial levels of k

and A may be set equal to the characteristic values that are obtained from equations

(7.3.1) to (7.3.5). On the other hand, if equations (7.3.6) and (7.3.7) are being used for

prescribing profiles ofk and A at the inlet boundary, intermediate values from these pro-

files can be used for the initial guess values.


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7.3.3 Solution Algorithms and Strategies

The type of solution approach (or strategy or algorithm) appropriate for any particular

turbulent flow problem depends on the following factors:

Physical attributes of the flow (for example, steady or transient, 2-D or 3-D)

Availability of computational resources (for example, memory limits may prohibit

you from using a faster but more memory intensive solution strategy)The following sections describe FIDAP solution algorithms and solution strategies for

turbulent flows. For a complete description of the solution algorithms available in FIDAP,

see Chapter 7 of the FIDAP Theory Manual.

Solution Algorithms

FIDAP provides two classes of algorithms for the numerical solution of the typical set of

discretized equations which result from the application of the Galerkin finite element

method to the governing flow equations.

Implicit algorithms

Explicit algorithm

Implicit Algorithms

The set of discretized algebraic equations that results from the stationary form of the flow

equations (or from the transient form of these equations when an implicit time integration

scheme is employed) is implicit for all flow variables. This means that non-trivial matrix

systems must be inverted to compute all the nodal unknown flow variables, such as

velocities, pressure, and temperatures. These matrix systems should be solved with one of

the implicit algorithms available in FIDAP.

The implicit algorithms are used for solving most flow problemswhether they are tur-

bulent or not. This is because most flows considered are either steady, or in the case of

transient flows, are integrated implicitly in timeprimarily to avoid stability limits on the

size of the time step, and/or to use the variable time stepping feature which is only avail-able with the implicit time integration schemes.

For the implicit algorithms, FIDAP provides two solution approaches for solving the

nonlinear equations that result from the finite element discretization:

Fully coupled approach

Segregated approach


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Fully Coupled Approach

In the fully coupledapproach, discretized equations are assembled into one large global

matrix equation system for later simultaneous solution. For the fully coupled approaches,

FIDAP allows the user some flexibility in designing the structure of the solution algo-

rithm. One may adopt either a mixedor a penalty formulation for the pressure-velocity

coupling. For nonlinear problems, two basic choices of linearization schemes are

available:

Fixed-point successive substitution scheme

One of the Newton-type schemes (that is, Newton-Raphson, Modified-Newton or

Quasi-Newton)

The global system of linear equations that results at each iteration from the fully coupled

approach is solved using direct LU factorization.

The fully coupled approach is the most direct solution approach used in FIDAP. Its great-

est advantage is that it typically takes the least number of iterations to arrive at a solution.

(The solution to linear flow problems is obtained in one iteration.) Moreover, this

approach is usually very efficient in terms of CPU time for most two-dimensional flows.

For very large, two-dimensional flows and most three-dimensional flows, however, the

CPU time requirements may become excessive. The main disadvantage of the fullycoupled approach is that its memory and disk storage requirements are the largest of all

solution approaches available in FIDAP.

Segregated Approach

In the segregated approach, the global matrix system is never directly constructed.

Instead, the discretized equations associated with each primary flow variable are assem-

bled in smaller matrix equation systems.

Unlike the fully coupled approach, the structure of the segregated approach is essentially

fixed and is not subject to user control. Thus, when this approach is selected, FIDAP

automatically adopts the mixed velocity-pressure formulation. Moreover, if the system of

flow equations is nonlinear, the fixed-point (or successive substitution) scheme is auto-

matically used to linearize the various nonlinear sub-matrix systems.

The default method employed in FIDAP for solving the various linear equation systems

resulting from the segregated solution approach is the direct LU factorization method. In

order to offset the high costs of the direct solver in large-scale 3-D problems, FIDAP also

provides the option of solving the above linear equation systems using iterative methods.

These solvers together with various preconditioning strategies are described in Chapter 7

of the FIDAP Theory Manual.


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The greatest advantage of the segregated approach is that it substantially reduces memory

and disk storage requirements compared to the fully coupled approach. This relative

resource requirement becomes more pronounced (in favor of the segregated approach) as

the number of transport equations in the flow model is increased. (This is especially true

when the iterative linear equation solvers are being employed.) However, because of its

decoupled nature, the segregated approach tends to require more iterations to produce a

solution than does the fully coupled approach. This is also true for linear flow problems,

where typically more than one iteration is required to obtain a solution. Our experienceindicates that, from the standpoint of CPU time, the segregated approach is not as effi-

cient as the fully coupled approach for most two-dimensional problems involving few

transport equations. However, the relative efficiency of the segregated approach improves

dramatically with an increase in the number of transport equations in the flow model. A

similar dramatic improvement is also gained when the dimensionality of the problem

increases from two to three. For example, for three-dimensional k- A simulations, the seg-

regated approach not only requires much less memory but also consumes substantially

less CPU time.

Explicit Algorithm

For the explicit algorithm, the set of discretized equations that results from the transient

form of the flow equations when the forward Euler time integration scheme is employedfor the temporal terms is explicit for all flow variables except the pressure. This means

that a non-trivial matrix system is encountered only for the pressure unknowns. The

matrix systems resulting for all the other flow variables are diagonal and are trivial to

invert. These matrix systems are solved as part of the explicit algorithm in FIDAP.

The explicit algorithm is automatically invoked in FIDAP when the forward Euler scheme

is used to integrate the transient flow equations in time. In contrast to the implicit algo-

rithms, which can be used for either the steady or transient flow equations, the explicit

algorithm is essentially a transient algorithm and is only appropriate for solving the time-

dependent version of the flow equations. Also, as is the case with the segregated algo-

rithm, the structure of the explicit algorithm is fixed within FIDAP and is not subject to

user control. This algorithm automatically uses a mixed velocity-pressure formulation,

and only requires the solution of one non-trivial matrix problem for the pressureunknowns. This system of linear equations for the pressure unknowns is solved using

direct LU factorization. (Iterative solvers may not be used here.)

An advantage of this algorithm is that it has a relatively small memory requirement. Its

main disadvantage is that the size of the time step is limited by various stringent stability

criteria. Thus, with this algorithm you typically need to use a relatively large number of

time steps to cover a time interval which is significant with respect to the characteristic

time scale of the mean flow process.


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Solution Strategies

The solution strategy is probably the most important factor in two-equation simulations of

turbulent flows. FIDAP solution strategies can be classified as belonging to one of two

general types.

Most 2-D flows

Large 2-D and most 3-D flows

Most 2-D Flows

For most 2-D flows in which the global system matrix can be accommodated in computer

memory and peripheral disk storage, the fully coupled implicit algorithm should be used,

because it tends to be the most efficient for such problems. Also, the penalty formulation

should usually be used with this approach, because it removes the pressure degree of free-

dom from the global system matrix. (In the penalty formulation, pressure is recovered in a

post-processing phase of the computation.) Moreover, the successive substitution scheme

should be used for linearizing the system matrix. The Newton-type schemes appear to be

inappropriate for the highly nonlinear turbulent flow equations resulting from the k- A

model and, for this reason, are disallowed for turbulent flows.

To invoke the fully coupled implicit algorithm in conjunction with the successive substi-tution scheme, you must specify the S.S. keyword on the SOLUTION command.

Large 2-D and Most 3-D Flows

For large 2-D flow problems and most 3-D problems, the segregated algorithm should be

used. It is invoked using the SEGREGATED keyword on the SOLUTION command. If

iterative linear equation solvers are to be employed, the recommended choice of solvers

and preconditioners is as follows.

Use the CR method for the symmetric pressure-type equation and the CGS method

for the non-symmetric advection-diffusion equations resulting from the various

conservation equations.

In conjunction with these solvers, use the SSOR preconditioners with the CRmethod and the diagonal preconditioners with the CGS method.

Use relatively tight tolerances (for example, 10 - 6) to ensure good overall conver-

gence behavior and solution quality.

The resulting SOLUTION command has the following keyword configuration:

SOLUTION(SEGR= n , CR= n , CGS= n , NCGC= 1E- 6, SGCG= 1E- 6, PRECON= 21)


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Implicit Algorithms

When using the implicit algorithms, it is very important to specify the relaxation of the

numerical solution in the iterative solution of the nonlinear turbulent flow equations.

Relaxation can be introduced in one of two ways:

The ACCF keyword of the SOLUTION command

The RELAXATION command

When the ACCF keyword is used, all flow equations present in the model are relaxed

equally and the relaxation factor which is used is equal to the value of the ACCF

keyword. With the fully coupled algorithm, this value may range from its default value of

zero for very simple unidirectional and confined flows to about 0.9 for highly complex

recirculating flows with strong intercoupling effects due to buoyancy or swirl or both.

However, a more typical range is between 0.4 and 0.7.

The RELAXATION command can be used to selectively relax the various flow

equations. This is the recommended approach for the segregated algorithm. The relaxation

factors typically used for this algorithm are about 0.1 to 0.4 for velocities, 0 to 0.2 for

pressure, 0 to 0.1 for temperature and species concentration, and 0.1 to 0.3 for k and A .

If the segregated algorithm is used and no relaxing is explicitly specified by means of the

ACCF keyword or the RELAXATION command, FIDAP automatically employs a set of

non-zero default relaxation factors for the various flow equations that are present in the

flow problem. (For more information, see the FIPREP Users Manual). Note that the

ACCF keyword takes precedence over the RELAXATION command. If this keyword is

present on the SOLUTION command and has been assigned a value, this value is the

relaxation factor that will be used for all flow equations. Any values entered using the

RELAXATION command are ignored.

Explicit Algorithm

The explicit algorithm should be used for those transient flow problems where very small

time steps are required to accurately resolve the temporal character of the flow. For prob-

lems of moderate size, the explicit algorithm may require the least CPU time per time

step. It should be noted, however, that cases requiring the use of very small time stepsseldom occur in practice. As a result, the explicit algorithm is rarely needed for turbulent-

flow computations.


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7.3.4 Mesh Density and Distribution

In general, a denser computational grid should be employed for a two-equation simulation

compared to the mesh of the corresponding laminar or zero-equation simulation. This is

due to the fact that, in a typical flow problem, the turbulence variables undergo much

sharper spatial variations and involve considerably more detailed features than do the

mean flow variables (that is, the velocity and temperature fields). Thus, if a grid-inde-

pendent solution to the mean flow field is to be obtained, the computational mesh must befine enough to resolve the details of the k and A (or M ) fields. Care must also be exer-

cised in the design of the computational mesh to ensure a smooth spatial distribution of

nodal points throughout the flow domain. Abrupt jumps in mesh density may lead to

spurious spatial oscillations in the flow variables, especially if they occur along the direc-

tion of the flow in regions where the grid Reynolds numbers are large. In extreme cases,

these so-called wiggles cause the numerical solution to diverge.

7.3.5 Solution Stability

Sources of Instability

There are three main sources of instability that, if untreated, adversely affect typical two-

equation simulations. Instability associated with the dissipation (or sink) terms in the k and A equations

Instability associated with the advection terms in the k and A equations.

Instability that ensues when the k- A model is used in the prediction of flows con-

taining both turbulent and laminar regions

These instabilities manifest themselves in terms of non-physical negative k and A (or M )

values (k, A , and M are physically positive definite quantities which can never assume

negative values in the real world) and/or highly unrealistic turbulence time and length

scales resulting from very small k and A values in flow regions where turbulence levels

have practically collapsed. The seat of these instabilities lies in the k and A (or M )

equations.

Instability Due to Dissipation Terms

The first source of instability is that associated with the dissipation (or sink) terms in the

k and A (or M ) equations. Physically, these terms act to maintain finite levels of k and A

(or M ). In the absence of these terms k and A levels grow uncontrollably (and exponen-

tially) due to the generation (or source) terms of the k and A (or M ) equations.


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During the course of a typical numerical solution, while the interim solution field is sig-

nificantly different from the fully converged solution, the dissipation terms may strongly

dominate the generation terms and can momentarily produce destabilizing negative nodal

values of k and A (or M ). These negative values are destabilizing, because they change

the polarity of crucially important processes in the k and A (or M ) equations. For

example, the turbulent viscosity will become negative causing highly destabilizing nega-

tive diffusion. The source and sink terms will change polarityfor example, the turbulent

generation terms, instead of extracting turbulence energy from the mean flow process,will extract energy from the turbulence field and impart it to the mean flow. Similarly the

sink terms, instead of dissipating turbulence energy, will generate turbulence energy (a

physical impossibility).

Instability Due to Advection Terms

The second source of instability is associated with the advection terms in the k and A (or

M ) equations. It is well known that, at large Reynolds numbers, these terms produce

stream-wise oscillations in the corresponding flow variables if they are approximated

using accurate non-diffusive discretization operators. (Such operators automatically result

from the application of the Galerkin finite element method to the flow equations.) Nega-

tive nodal values of k and A (or M ) could therefore result if these oscillations are large

compared to the local values ofk and A (or M ).

Instability in Flows with Both Laminar and Turbulent Regions

The third source of instability ensues when a two-equation model is used in the prediction

of flows containing both turbulent and laminar regions. A typical example of this is in ex-

ternal aerodynamic problems where the free-stream flow is turbulence-free and the flow

surrounding the body is fully turbulent. The k and A equations of the standard high-

Reynolds-number k- A turbulence model become anomalous in the turbulence-free

regions. These equations contain terms involving ratios between k and A (that is, k2

A ,

k A , A k and A2

k) which are clearly indeterminate in laminar flow regions where turbu-

lence is not present (that is, k and A are zero). Numerically, these ratios become very sen-

sitive to noise level variations in k and A and begin to oscillate violently (and

anomalously) from one nodal point to another. This can have a devastating effect on the

numerical stability of the computation, as this unstable behavior quickly spreads to the

fully turbulent flow regions and in a matter of a few iterations totally contaminates the nu-

merical solution.


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Stabilization Techniques

Two basic techniques are available in FIDAP for suppressing the instabilities described

above.

Upwinding

Clipping

Upwinding

The purpose of upwinding is to stabilize advection terms in flow regions of high

advection. FIDAP provides three types of upwinding:

Streamline

First-order

Hybrid

Streamline upwinding stabilizes the high-order symmetric advection operators that arise

in the Galerkin method by explicitly adding numerical diffusion only along the flow

direction. In doing so, it helps to suppress the stream-wise oscillations in the various flow

variables (including k andA

) that occur in advection dominated flow regions. In first-orderupwinding, the convection operator is decomposed in the principal directions, then

full upwind differencing is applied in the respective principal directions. Hybridupwind-

ing is an extension of first-order upwinding in which a blending parameter ( = ) that is

used in the first-order scheme is dynamically computed during the course of the computa-

tions rather than being fixed by the user. (For detailed descriptions of the three upwinding

schemes, see Chapter 7 of the FIDAP Theory Manual.)

Streamline upwinding is the recommended upwinding technique for most FIDAP

problems. When you specify a problem as TURBULENT, FIDAP activates the streamline

upwinding scheme and employs default upwind factors of unity (1).

There are two ways to manually invoke streamline upwinding in FIDAP.

The UPWINDING keyword on the OPTIONS command The UPWINDING keyword on the OPTIONS command in conjunction with the

UPWINDING command

When the UPWINDING keyword on the OPTIONS command is used alone, all flow

equations present in the flow model are upwinded by equal amounts, and the upwinding

factor used is equal to the value of the UPWINDING keyword. A typical value for the

upwinding factor for turbulent flows is unity for k- A simulations, because they are

usually advection dominated. Lower values may be used if advection effects are not


27/109



strong. In extreme cases, values larger than one could be used. However, this value should

not exceed 2.

The UPWINDING command in addition to an OPTIONS(UPWINDING) command with

no keyword value may be used to selectively upwind the various equations present in the

flow model. The upwinding factors typically used with this command for turbulent flows

are about unity for the mean flow variables (velocities, temperature and species concen-

tration) and between one to five for k and A . Note that an UPWINDING keyword value

on the OPTIONS command takes precedence over the UPWINDING command. If this

keyword value is present on the OPTIONS command, the upwinding factors assigned via

this keyword will be used for all flow equations and any such values entered using the

UPWINDING command are ignored.

Clipping

Clipping is a procedure that avoids the first and third types of instability described above

by ensuring that nodal values of k and A do not fall below pre-assigned lower bound

positive values. This procedure is described under the VISCOSITY command in the

FIPREP Users Manual.

The lower bound values below which nodal values of k and A are clipped are set by

default to be ten million times smaller than the maximum nodal values ofk and A . This isadequate for virtually all cases. However, in flows involving a combination of exception-

ally large mean flow Reynolds numbers (that is, Re > 107) and solid walls, or in flows

involving a wide range of physical scales (such as a jet issuing into a large chamber), the

lower bound values at which clipping occurs must be lowered. This is done using the

CLIP keyword on the VISCOSITY command by assigning a value to this keyword that is

greater than 107.

Note that streamline upwinding and clipping should be regarded as being artificial stabil-

ity-enhancing measures, because they interfere with the course of the numerical solution.

This is especially true of streamline upwinding. If used correctly, the amount of upwind-

ing used must be just enough to suppress anomalous stream-wise oscillations. Excessive

upwinding (accomplished using larger than required upwinding factors) can lead to false

numerical diffusion in the stream-wise direction which can in turn modify the truenumerical solution. The default upwinding factor employed in FIDAP should be optimal

for most simulations.


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Checklists MODELING TURBULENT FLOWS


7.4 Checklists

7.4.1 Turbulence Modeling Checklist

The following is a checklist of the steps specific to a simulation using a turbulence model:

1. Specify the TURBULENT keyword on the FIPREP PROBLEM command.

2. Specify any turbulent constants or specialized options using the TURBOPTIONScommand.

3. Select the turbulence model desired with the MIXLENGTH, MIXING, or TWO-

EQUATION keywords on the VISCOSITY command.

a) If you specify the MIXLENGTH model, you must supply the USRMXL sub-

routine and create an executable module. (See Appendix C of the FIDAP

Users Manual for a general discussion of user subroutines.)

b) If you specify the TWO-EQUATION model:

i) Specify a two-equation model by means of the TURBOPTIONS

command. (The default STANDARD option indicates the use of the stan-

dard k-A

model.)ii) Specify an eddy viscosity constitutive relation by means of the EDDYVIS-

COSITY command. (The default BOUSSINESQ option indicates the use

of the Boussinesq eddy viscosity constitutive relation.)

iii) Specify any desired constrained values for k and A (or M ) at the inlet and

entrainment boundaries using the KINETIC and DISSIPATION keywords,

respectively, on the BCNODE command. (Specification ofk and A (or M )

is not required on wall boundaries.)

iv) Specify any desired initial values for both k and A (or M ) using the

KINETIC and DISSIPATION keywords, respectively, on the ICNODE

command. This step is strongly recommended.

4. For any solid wall boundaries, specify WALL boundary elements (see Section

7.4.2, below).


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MODELING TURBULENT FLOWS Checklists


7.4.2 Wall Boundary Elements Checklist and Recommendations

The following sections present a checklist for the specification of wall boundary elements

and recommendations that apply to the following flow types and turbulence models:

High-Reynolds number (high-Re) k- A models

Low-Reynolds number (low-Re) k- M models

Wall Boundary Elements Checklist

The following is a checklist of the steps particular to the specification of wall boundary

elements for turbulent simulations:

1. Define a WALL boundary entity on any wall boundaries of the model.

2. Create the mesh:

a) For high-Re k- A models, create the mesh such that the first layer of elements

is thick enough to completely contain the viscous sublayer and transition

region in the near wall region.

b) For low-Re k- M models, use enough grid points in the near-wall regions to

accurately resolve the sharp profiles of the mean flow variables as well as the

turbulence variables.

3. Set velocity components on wall boundaries just as they would be set for non-

turbulent analysesthat is, fixed walls are defined with all velocity components

set to zero and moving walls have the appropriate velocity component set to a

specified velocity.

4. Input constants affecting the near-wall modeling on the TURBOPTIONS

command. (In almost all cases the defaults should be satisfactory.)

5. Nodes on a WALL boundary do not require the specification of any kinetic energy

or dissipation boundary conditions.


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Checklists MODELING TURBULENT FLOWS


Recommendations

High-Re Models

After the solution, to assure that the first layer of elements is thick enough to completely

contain the viscous sublayer and transition region, use the YPLUS command in FIPOST

to plot the y ++ values at the WALL boundaries. (The y ++ value is a dimensionless distance

related to the various layers present in a turbulent flow. You can have y

++

values formomentum, temperatures and species, depending on the set of equations you are solving.)

If the value of y+

+ for the momentum layer is greater than 30 for all elements then these

elements are thick enough. Values of y+

+ lower than 30 for the momentum layer may be

safely tolerated provided these are not occurring in wall regions of crucial importance to

the overall flow process.

In isothermal flows, the predicted velocity field is generally insensitive to y+

+ values in the

range 10 1000


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MODELING TURBULENT FLOWS Checklists


made available for plotting in FIPOST to shed more light on the underlying physical

processes. For example, in gaseous flows where the laminar Prandtl number is of the

order of unity, the thermal y ++ values will be close to the momentum y ++ values indicating

that the viscous and thermal sublayers have similar thicknesses. In high Prandtl number

fluids such as oils where the thermal sublayer is much thinner than the momentum

sublayer, the thermal y+

+ value will be much larger than the momentum y+

+ values.

Conversely, in low Prandtl number fluids such as liquid metals, where due to the

enhanced thermal diffusion the thermal sublayers are very thick, the thermal y+

+

valueswill be much smaller than those of the momentum y ++ values. Similar considerations apply

to the relative thicknesses of the species and momentum sublayers and their correspond-

ing y+

+ values where the relevant dimensionless number representing the relative impor-

tance of momentum to species diffusion is the laminar Schmidt number.

Low-Re Models

For low-Re k-w

simulations, it is not possible to check the y ++ values in FIPOST (as can

be done for the special elements while using the k- e models). The best way to insure that

the near-wall mesh is sufficiently fine is to compare the value of the molecular viscosity

to the value of the turbulent viscosity, m t, on nodes adjacent to the wall. The value of m t

must not exceedm

in those regions. Ifm

t exceedsm

, a finer mesh should be used in thenear-wall region.


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MODELING COMPRESSIBLE FLOWS Preliminary Remarks


8. MODELING COMPRESSIBLE FLOWS

FIDAP provides extensive capabilities for the simulation of compressible flows. This

chapter summarizes these capabilities and the steps required to set up and run a com-

pressible flow simulation using FIDAP. More complete details on the theoretical under-

pinnings of both the equations solved and the numerical techniques employed can be

found in the FIDAP Theory Manual.

8.1 Preliminary Remarks

As a prerequisite to reading this chapter you are strongly recommended to read Chapter 2

of the FIDAP Theory Manual which describes in detail the various flow governing equa-

tions. One such equation is the equation of state (EOS) which in its most general form

relates density, temperature, pressure and species concentrations. This relationship may be

written as;

f p T c c( , , , , , ...)H 1 2 0= . (8.1.1)

In FIDAP the equation of state is used to define density in terms of other flow variables

and is specified via the fluid density property model. The density model is input using the

DENSITY command which is then associated with a particular entity (for example, afluid) via the ENTITY command.

Various types of density models may be employed in FIDAP ranging from constant

density models to models where the density is a function of pressure, temperature and

species concentration. The density model also determines whether the flow process is to

be considered as being incompressible or compressible. This is a very important distinc-

tion as it determines the exact form of flow equations that will be employed in FIDAP and

the choice of algorithms that are available for solving these equations. As noted above,

this chapter is mainly intended for describing the compressible flow capabilities of

FIDAP. However, before this is done a brief description of incompressible flow capabili-

ties is given in the following section.


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Incompressible Flows MODELING COMPRESSIBLE FLOWS


8.2 Incompressible Flows

The density variation in many flow processes of practical interest is negligibly small.

Examples of these are flow processes involving liquids such as water, oils, molten metals,

etc. Other examples are flow processes involving gases subject to moderate pressure

and/or temperature gradients. Such flows can be accurately modeled by assuming the

density to be a constant. The appropriate EOS is therefore an equation of the type,

H H

= 0 (8.2.1)

which is the simplest form of equation (8.1.1). Such an equation of state is specified in

FIDAP via the DENSITY command using the CONSTANT model, where the value of

the CONSTANT keyword is the value of H 0 . An incompressible flow is specified using

the INCOMPRESSIBLE keyword on the PROBLEM command; in this case, all density

models employed in a flow simulation must be of the CONSTANT type. Note that in an

incompressible flow simulation the density need not be the same everywhere in the com-

putational domain. An example of this is a flow problem involving two fluids (say, air and

water) separated by a solid region. Another example is a flow problem involving two

immiscible fluids (say, oil and water) separated by a sharp internal interface. In both these

examples the density of each fluid is input using a CONSTANT density model.

The most important ramification of an INCOMPRESSIBLE flow simulation is that thecontinuity equation assumes the following simplified form:

=u 0 (8.2.2)

which is often referred to as the incompressibility constraint. The density appearing in the

remainder of the flow governing equations assumes its prescribed CONSTANT value.

The only exception to this is in flows involving buoyancy forces (the BUOYANCY key-

word specified on the PROBLEM command) where the Boussinesq assumption is

employed to model the body force term ( )H H- 0 g which appears in the momentum

equation. For this term only, the CONSTANT density model allows two representations

of the density variation which may result from temperature and/or species concentration

variationsbut not pressure variations. These two alternative representations may be

selected by specifying the TYP1 or TYP2 keywords of the DENSITY command. Formore details on the exact forms of the buoyancy term refer to Chapter 9 of the FIPREP

Users Manual where the DENSITY command is discussed in detail.


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MODELING COMPRESSIBLE FLOWS Compressible Flows


8.3 Compressible Flows

There are also many flow processes encountered in practice where the fluid density varies

significantly. This typically occurs in flow processes involving gases where the pressure

and/or temperature variations are large and of the order of the absolute pressure and/or

temperature levels. Significant density variations may also result in gaseous mixtures

involving reacting chemical species provided that the chemical reactions produce signifi-

cant changes in the gas constant of the mixture. Noticeable density variations may alsooccur in liquids. However, these are not as large as typical density variations encountered

in gases and are due primarily to temperature and/or species concentration variations.

Finally, small density changes may also occur in liquids as a result of extreme pressure

variations, but these situations are not as common.

A large number of equations of state exist which approximate the dependence of density

on the other primary flow variables (see equation (8.1.1)). Some of these are more general

and can be applied to a large class of fluids under a relatively large range of working con-

ditions. Others are more specialized, intended for specific fluids operating under a

narrow, and often extreme, range of working conditions. FIDAP offers three main non-

constant categories of density models from which the user may construct a relatively wide

range of equations of state. These are:

Variable density model

Ideal gas law density model

User-specified density model entered by means of a user-supplied subroutine

These density models are respectively activated via the VARIABLE, IDEAL and SUB-

ROUTINE keywords of the DENSITY command. Before these non-constant density

models are discussed in detail it is necessary to introduce a number of definitions, con-

ventions and notions that are of fundamental importance to FIDAP and its proper opera-

tion. While not all of these are unique to the modeling of compressible flows, they are

particularly relevant to such simulations.

8.3.1 Definitions and Conventions

This section defines the types of flows that FIDAP considers as being compressible. It

provides definitions for single- and multi-component flows, and introduces the notion of

the carrier fluidas it relates to single- and multi-component fluids.

When a compressible flow is specified by the COMPRESSIBLE keyword on the PROB-

LEM command, then at least one of the density models input must be of the non-constant

type (that is, VARIABLE, IDEAL or SUBROUTINE). It is also very important to note

that there is a fundamental difference in the types of equations of state that can be con-

structed with the above three types of density models. The VARIABLE density model


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Compressible Flows MODELING COMPRESSIBLE FLOWS


allows the density to be a function of temperature and/or species concentration, but does

not allow any dependence on pressure. In contrast, the IDEAL and SUBROUTINE

density models require as a minimum that density be a function of both pressure and tem-

perature; if chemical species are also present in the flow process, you may optionally

choose to make density a further function of species concentrations. When density is

allowed to be a function of pressure, the physical flow process becomes more intricate

and intercoupled. This gives rise to phenomena that would not otherwise be present.

Probably the most significant of these is the phenomenon of sound wave propagation. Animportant consequence of this is that shocks may be produced in flow regions where the

local speed of flow exceeds the local speed of sound. Since the flow processes resulting

from the VARIABLE and IDEAL/SUBROUTINE density models are fundamentally dif-

ferent, FIDAP does not allow the mixing of these two classes of density models in the

same simulation. Thus, a COMPRESSIBLE simulation may be performed with various

density models of the VARIABLE type. Alternatively a COMPRESSIBLE simulation

may be performed with various density models of the IDEAL and/or SUBROUTINE

type. However, such a simulation may not be performed involving a VARIABLE density

model and an IDEAL (or SUBROUTINE) density model concurrently. Because of these

restrictions, the form of the energy equation adopted in FIDAP in a COMPRESSIBLE

flow simulation will depend on the particular density model employed. Additionally, if the

simulation is time-dependent (TRANSIENT keyword on the PROBLEM command), the

form of the continuity equation will depend on the solution algorithm employed. Specificdetails are provided later in the following sections where the above three density models

are more fully described.

The notion of the carrier fluid is of fundamental importance to FIDAP and has a some-

what different connotation depending on whether a single- or multi-component simulation

is being performed. A single-component simulation comprises a single homogeneous

fluid. Examples of such fluids are pure substances such as water or nitrogen gas. How-

ever, in the context of FIDAP, non-pure substances may also qualify as being single-com-

ponent fluids provided they are well mixed. The best known example of this is air which

is composed of a homogeneous mixture of pure gases. In a single-component simulation

the working fluid is the carrier fluid. This means that the various flow governing equa-

tions of momentum, continuity, temperature, etc. (no species equations may of course be

present in this context), and all property models defined in FIDAP (that is, for specificheats, conductivities, etc., as well as densities), pertain to the carrier fluid.

A multi-componentsimulation comprises fluids involving a non-homogeneous mixture of

more that one substance. The convention used in FIDAP to determine the number of

components N present in the mixture is as follows. The number of components is equal to

the number of species equations present in the flow, plus onethe carrier fluid. For

example, in a flow simulation involving a three-component mixture (N= 3), only two

transport equations need to be solved (for c1 and c2 )the species concentrations of any

two of the three mixture components. The third component is regarded as being the carrier


36/109



fluid which is arbitrarily chosen to be any one of the three components of the mixture. In

a multi-component mixture the various flow governing equations (momentum, continuity,

temperature, species concentration, etc.) and all property models defined in FIDAP

pertain to the mixture. Moreover, the principle of conservation of matter dictates the

following relationship between the species concentration cN of the carrier fluid and the

species concentrations of the remaining N- 1 species components;

c cN nn

N

= -

=

-

11

1

. (8.3.1)

This relationship is inherent in all the non-constant density models available in FIDAP

and provides the crucial link which relates the density of the mixture to the mass fractions

of the various chemical components present in the flow. It is important to note that in its

strictest sense the summation in equation (8.3.1) involves all the N- 1 chemical species

present in the flow simulation as it is assumed that all these will contribute to the density

of the mixture. However, in practice this is not always the case as there are situations

where some of the species components may be neutral (or near neutral) from the stand-

point of the overall mass of the mixture. Examples of these are tracer fluids which may or

may not be chemically inert. For this reason when a non-constant DENSITY model in

entered in FIDAP, you are required to explicitly specify which chemical species contrib-

ute to the summation in equation (8.3.1). The manner in which this is done is explained inthe following section.

8.3.2 The VARIABLE Density Model

The VARIABLE density model allows the density to be represented in the general form:

H

H >

=

+

0 0

1

1

1

1 1

- -

-

FHG

IKJ

=

-

T

N

n

n

n

N

T T

M

Mc

b g.

(8.3.2)

The following approximate version of equation (8.3.2) is also available which is referred

to as the product form:

H H > >

= - - -

-

0 1 101

T c n

n

N

T T cn

b g d i . (8.3.3)

The above forms of the VARIABLE density model are invoked via the TYP1 or TYP2

keywords of the DENSITY command. TYP1, which is the default form, selects equation

(8.3.2) and TYP2 selects equation (8.3.3). In the above equation, H 0 is the reference

density which is the value of the VARIABLE keyword. > T and > c are respectively, the

coefficients of volume expansion due to temperature and species concentration, and T0 is

a reference temperature. The value of T0 and the property models for > T and > c are


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entered via the VOLUMEXPANSION command (see Chapter 9 of the FIPREP Users

Manual for details). Mn is the molecular weight of the n th chemical species, and MN is

the molecular weight of the carrier fluid. The molecular weights are entered via the

MOLECULARWEIGHTS command (see Chapter 9 of the FIPREP Users Manual for

details).

Recall that in setting up a density model, you are required to specify explicitly which of

the species components present in the simulation will be contributing to the summationspresent in the above equations. How this is done is demonstrated by way of the following

example. Consider a six-component COMPRESSIBLE flow simulation involving the five

species, c1, c2 , c3 , c4 , and c5 (that is, keywords SPECIES = 1, SPECIES= 2, ..., SPECIES

= 5 specified on the PROBLEM command), where you wish to define a density model

making density a function of the three species components c2 , c3 , and c5 only. The

DENSITY command for that particular model must include the three keywords,

SPECIES = 2, SPECIES= 3 and SPECIES= 5. If these keywords are not specified the

density will not be a function of species concentration.

Equations (8.3.2) and (8.3.3) above may be manipulated to create a wide range of equa-

tions of state for density.

In the case of a single-component fluid (carrier fluid only), the VARIABLE model canonly be used if the temperature equation is present (ENERGY or SPECIES = 0 keyword on

PROBLEM command). In this context the form of equations (8.3.2) and (8.3.3) are the

same and is as follows:

H H >

= - -

0 01 T T Tb g . (8.3.4)

In the case of a multi-component fluid, if the temperature equation is not present (or if it

is present but the TEMPERATURE keyword is not specified on the DENSITY com-

mand) the VARIABLE density model assumes different forms depending on whether the

TYP1 or TYP2 keywords are specified. If TYP1 is specified, the following form is

employed:

H

H

=

+ -

FHG

IKJ

=

-

0

1

1

1 1M

McN

n

n

n

N . (8.3.5)

If TYP2 is specified the density equation becomes:


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H H >

= -

-

01

1 c nn

N

ncd i . (8.3.6)

Again remember that it is your responsibility to ensure which species components are

activated in the above expressions by selecting the appropriate SPECIES = n keywords on

the DENSITY command.


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NOTES

1. The VARIABLE and IDEAL (or SUBROUTINE) density models cannot be used

together in a COMPRESSIBLE simulation.

2. In a COMPRESSIBLE flow simulation involving VARIABLE type density mod-

els, the form of the temperature equation employed in FIDAP is as follows:

H

c Tt

u Tx

qx

Hp jj

j

j

+FHG IKJ= - + (8.3.7)

where cp is the specific heat at constant pressure, qj is the flux of thermal energy

andH is the general heat generation/destruction term.

3. The form of the continuity equation in TRANSIENT flows will depend on the

type of solution algorithm employed. If a SEGREGATED solution algorithm is

employed, the full form of the continuity equation is employedthat is,

r

r

t

u

x

j

j

+ =

c h0 . (8.3.8)

If one of the fully coupled solution algorithms is employed (that is, either one ofthe S.S., N.R., Q.N. or M.N. keywords on the SOLUTION command), the fol-

lowing truncated form of the continuity equation is employed:

r

u

x

j

j

c h= 0 . (8.3.9)

Moreover, if a fully coupled solution algorithm is employed, only a MIXED pres-

sure-velocity formulation is allowedthe PENALTY formulation may not be

employed. Note finally that a TRANSIENT COMPRESSIBLE simulation may not

be performed with the explicit FORWARD Euler solution algorithm.


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8.3.3 The IDEAL and SUBROUTINE Density Models

Recall from above that the IDEAL and SUBROUTINE density models require that den-

sity is at least a function of temperature and pressure. This means that in COMPRESSI-

BLE flow simulations involving IDEAL and/or SUBROUTINE density models at least

the momentum, continuity and temperature equations mustbe present (the ENERGY (or

SPECIES= 0) and MOMENTUM keywords specified on the PROBLEM command).

The IDEAL density model activates the well-known ideal gas law equation of:

H

=

+ -

FHG

IKJ

L

NM

O

QP

=

-

M p

R TM

Mc

N

N

n

n

n

N

1 11

1.

(8.3.10)

In the above equation,p and Tare the absolute thermodynamic pressure and temperature

respectively, and R is the universal gas constant. The value of the universal gas constant

R may be entered in one of two ways. A unique constant value may be entered for each

fluid ENTITY present in the computational domain via the GASCONSTANT keyword on

the DENSITY command. If, on the other hand, R is constant throughout the computa-

tional domain (which is usually the case), its value may be entered as a global parameter

via the GASCONSTANT keyword on the COMPRESSIBLE command. It is important tonote that the global value of R entered using the GASCONSTANT keyword of the

COMPRESSIBLE command will be overridden by any R value entered via the GAS-

CONSTANT keyword of the DENSITY command.

Note the similarity between the above equation and equation (8.3.2) of the VARIABLE

density model which has been derived from the ideal gas law model. The significant dif-

ference between them is the appearance of pressure in equation (8.3.10). A further differ-

ence is that the temperature appearing in the VARIABLE density model need not

necessarily be the absolute thermodynamic temperatureit may be the temperature per-

turbation about a reference temperature T0 .

Finally, as is the case in the VARIABLE density models, it is your responsibility to en-

sure which species components are active in a given IDEAL density model by selecting

the appropriate SPECIES = n keyword(s) on the DENSITY command.

The SUBROUTINE density model allows you to construct ideal-gas-law-type equa-

tions of state of the form:

1

H

=

FHG

IKJf

p

T. (8.3.11)


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For this option, you must supply a subroutine USREOS that returns the value of the

density at any point in the fluid based on such an equation of state. The format of the

USREOS subroutine is presented in Chapter 9 of the FIPREP Users Manual where the

DENSITY command in described. Although a great deal of flexibility for constructing

customized equations of state is provided by the USREOS subroutine, the equation of

state constructed in this manner must satisfy the functional form of equation (8.3.11).

NOTES

1. As noted above, the phenomenon of sound wave propagation is present in COM-

PRESSIBLE flow simulations employing the IDEAL and/or SUBROUTINE den-

sity models. This means that shock waves may potentially be formed in the flow if

the fluid speed is large. FIDAP is only intended for flows where the fluid speed is

lower than the speed of soundthat is subsonic flows where the Mach number in

the flow is everywhere less that one (Ma < 1).

2. In a COMPRESSIBLE flow simulation employing IDEAL and/or SUBROUTINE

density models, the form of the energy equation solved in FIDAP is:

H

cT

t

uT

x

q

x

H pu

x

v j

j

j

j

j

j

+

F

HG

I

KJ = - + - Ec (8.3.12)

where cv is the specific heat at constant volume whose value is computed in each

fluid ENTITY from the following equation:

c cR

M

M

Mcv p

N

N

n

n

n

N

= - + -

FHG

IKJ

L

NM

O

QP

=

-

1 11

1

. (8.3.13)

Note that no additional property model need be entered for computing cK

as all

information necessary for its evaluation is already available through the property

models entered for the fluid entity for cp , R and the molecular weights, Mn and

MN . Also, the species components active in the summation in equation (8.3.13)

are the same as those that have been activated in the DENSITY model entered for

the fluid entity.

The last term in equation (8.3.12) is the reversible pressure work term which

causes significant temperature changes in flows where Ma > 0.3. The dimension-

less coefficient Ec multiplying the pressure work term is provided to facilitate the

nondimensionalization of the energy equation. This coefficient is assigned to a

particular fluid entity via the REVERSIBLE keyword of the ENTITY command.

If the flow problem is solved in dimensional form, the value of the coefficient Ec

is unity which is the default value of the REVERSIBLE keyword. However, if the

flow problem is formulated and solved in dimensionless form, the value of Ec


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may not be unity, and therefore needs to be explicitly entered using the

REVERSIBLE keyword.

In a COMPRESSIBLE simulation using the IDEAL and/or SUBROUTINE den-

sity models, the flow problem must always be set up in terms of absolute tem-

peratures. It is strongly recommended that a non-zero initial (or guess) temperature

field be prescribed everywhere in the computational domain. This can be done

through the use of the ICNODE(TEMP, CONSTANT = v , ALL) command.

3. The correct modeling of pressure is also of crucial importance in flows employing

the IDEAL and/or SUBROUTINE density models. In these simulations the level

of absolute pressure must be correctly imposed and strictly controlled. (This con-

trasts with flow simulations employing CONSTANT or VARIABLE density mod-

els where the absolute level of pressure is unimportant and it is only necessary to

model pressure variations with respect to an arbitrarily set pressure datum.) The

absolute pressure level may be set by a combination of the normal stress boundary

condition applied at an outflow boundary and a reference pressure level value

imposed through the PRESSURE keyword of the COMPRESSIBLE command. It

is strongly recommended that a non-zero reference pressure level be set. All com-

puted pressures in FIDAP will then be perturbations about this referenc

6819874 Turbulent Flows

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Transcript of 6819874 Turbulent Flows