2024年3月16日发(作者：)

本文主要介绍ANSYS CFX 11.0中多孔介质模型的使用方法。首先详细讲述了

Porosity Settings 对话框的填写方法，之后以附录形式给出了多孔介质模型中的定

义、术语、方程等供参考。以下内容为本人根据CFX帮助及相关资料编写，错漏之

处敬请见谅并指正。

Porosity Settings 对话框填写说明

Porosity Settings对话框包括三项：Area Porosity、Volume Porosity和Loss Models。

1. Area Porosity： 即面积孔隙率，是指流体可以穿过的面积占物理面积的份额，默

认为Isotropic（各向同性的），不能修改。

2. Volume Porosity：即体积孔隙率，是指允许流体流动的体积与物理体积之比。

3. Loss Models： 即阻力损失模型，可选择Isotropic Loss（各向同性）或Directional

Loss（各向异性）。此外还有多项需要选择或填写：

3.1 Loss Velocity Type：即阻力损失对应的速度类型。可选择Superficial（表观

流速，即按物理面积计算的流速）或True Velocity（真实流速）。

3.2 若选择了Isotropic Loss，则需要填写Isotropic Loss对话框，其界面如图1：

图1 Isotropic Loss对话框

在Option（选项栏）中，有两种阻力计算方式可以选择：

3.2.1 Permeability and Loss Coefficient（渗透率和损失系数）

分别填写渗透率和损失系数即可。

渗透率为多孔介质本身的性质，需通过试验测定，计算式为：

K

=

Q

μ

L

AΔP

其中，Q为通过多孔介质的体积流量，μ为动力粘度，L为流通长度，A为

横截面积，ΔP为压差。由该式可知，渗透率具有面积的量纲。其常用单位

为达西（Darcy），物理意义为：介质允许粘度为1cp的流体，在压力梯度

为1atm/cm的作用下，通过横截面积为1cm

2

的流量为1cm

3

/s，此时，介质

的渗透率称为1达西。换算关系为：1Darcy =1 μm

2

=10

-12

m

2

。

损失系数也是一个试验系数，它表示了流道内摩擦阻力损失与局部阻

力损失之和，其计算式可由下式推出：

L

ρ

v

2

ρ

v

2

ρ

v

2

ΔP=ΔP+

ζ

=K

loss

L

1

+ΔP

2

=

ξ

22

D

e

2

为：

K

loss

=

ξ

D

e

+

ζ

L

其中，

ξ

为摩擦阻力系数，

ζ

为局部阻力系数，

De

为当量直径，

L

为流通长度。

实际应用中可以通过阻力与速度的试验关系直接得到损失系数，而不需要区

分摩阻和形阻等。

3.2.2

Linear and Quadratic Resistance Coefficients

（线性的和平方的阻力系数）

分别填写线性阻力系数和平方阻力系数即可。这两个系数可由渗透率和

损失系数计算出，

分别为：

C

R

1

=

μ

K

，C

R

2

=K

loss

ρ

2

不论选择哪种阻力计算方式，都应注意：试验数据中使用的速度应当与计算

中使用的速度一致。若试验数据中使用的是真实速度，而计算中使用的是表观速

度，则渗透率和损失系数应当调整为：γK和K

loss

/γ

2

；而线性的和平方的阻力系数

应当调整为：

C

R

1

=

μρ

，C

R

2

=K

loss

2

γ

K

2

γ

其中，γ为体积孔隙率。

3.3

若选择了

Directional Loss

，则需要填写Directional Loss对话框，界面如图2：

由图可见，本对话框又分为3个子对话框：Streamwise Direction（流动方向）、

Streamwise Loss（流向损失）、 Transverse Loss（横向损失）。

3.3.1

Streamwise Direction：在Option（选项栏）中指定坐标系，可选择

Cartesian Components（笛卡尔坐标系）、 Cylindrical Components（圆柱坐

标系），在选定坐标系下填写对应方向矢量即可。笛卡尔坐标系下为X、Y、

Z；圆柱坐标系下为Axial(轴向)、r(径向)、theta(角度)，同时需定义轴向，

可用已有坐标轴或两个点定义。

3.3.2

Streamwise Loss：有三种阻力计算方式，前两种与3.2节相同；第三种

为 No Loss（无损失），即忽略该方向的阻力损失。

图2 Directional Loss对话框

3.3.3

Transverse Loss：除可以使用3.3.2节的三种阻力计算方式外，还有一

种方式可以选择：Streamwise Coefficients Multiplier（轴向系数倍率）：指

定一个Multiplier（倍数），典型值为10-100；横向损失中的各系数（不包

括渗透率）由轴向指定的系数乘以Multiplier得到。

附：多孔介质中的流动

在ANSYS CFX中，多孔介质中的流动可以使用动量损失模型或多孔介质模型进

行计算。动量损失模型可以在流体域中使用，而多孔损失模型只能在多孔介质域中

使用。以下仅介绍多孔介质模型。

多孔介质模型基于纳维

-

斯托克斯方程和达西定律。它能够模拟那些几何结构太

复杂以致不便划分网格的问题。模型保留了对流项和扩散项，因此能够应用于这两

项的影响很重要的棒束或管束中的流动。

根据连续性方程，模型假设多孔介质中“无穷小”的控制体和控制面相对于孔

间隙来说仍然是很大的（尽管实际上它们可能小于孔隙的尺寸），因此，每个给定

的控制单元和控制面都既包含了固体域，也包含了流体域。

先介绍两个基本概念：

1

）体积孔隙率：一个点处的体积孔隙率

γ

是指该点附近的一个无穷小的控制

单元内允许流体流动的体积

V’

与物理体积

V

之比。即：

V'=

γ

V

；

2

）面孔隙率：假设贯穿无穷小平控制面

A

的允许流体流动的面积

A’

为：

A’=K

·

A

，

则其中的

K

叫做面孔隙率张量，它是一个对称的二阶张量。目前，ANSYS CFX 仅

允许各向同性的面孔隙率张量

K

。

下面首先介绍达西定律。该定律是1856年法国水利工程师达西为解决水的净化

问题从大量实验中总结出来的。达西对水通过均匀砂层的缓慢流动作了大量实验,

研究表明：单位时间流过砂层的体积流量Q与横截面积A、测压管水头差h1-h2成

正比,与流过的砂层长度L成反比，即：

Q

=

KA

h

1

−h

2

L

定义Q/A=v为渗流速度，(h

1

-h

2

)/L=J为水力坡度，则上式也可写成：

v＝kJ

这就是达西定律。式中，k为标志渗流能力大小的实验常数，称为渗透系数。它既

与砂层的结构有关，又与流过的流体性质有关。由量纲分析知：k=Kρg/μ,其中,ρ、

μ分别为流体的密度和动力粘度，g为重力加速度，K为介质的渗透率。将该式代入，

则达西定律也可表示为：

v

=

K

ρ

g

μ

J或J=

μ

v

K

ρ

g

实验发现,随着雷诺数Re的增加,多孔介质中的流动状态经历三个区域：

①线性层流区：粘性力占优势，达西定律成立,上限约在Re=10左右。

②非线性层流区（过渡区）：为主要被惯性力制约的层流，达西定律不成立。

此时，流体与介质间的表面分子力作用显得更为重要，部分液体的滞流现象使孔隙

率发生变化，从而引起渗透率的相应变化。实验表明，这时孔隙率和渗透率均随渗

流速度的增加而增加，速度到某一临界值后不再变化，因此不遵循达西定律。非线

性渗流定律的一般形式可写为：

v＝kJf(J)

式中，f(J)为小雷诺数情况下渗透率随水力坡度的变化函数关系，由实验确定。

该区上限约在Re=100左右，在上限附近开始有层流到湍流的过渡；

③湍流区：惯性力占优势，达西定律不成立。 此时应该用“渗流的二项式定律”

代替达西定律，即：

J＝Av＋Bv

2

式中A、B为决定于流体和介质性质的常数。

以上是单相流体达西定律；对于多相流体，达西定律对每一相仍然成立，只需

将渗透率修正为该相的相渗透率即可。

在CFX中，使用压力梯度代替无量纲量水力坡度J，故达西定律变为：

−

dpΔp

ρ

g

(

h

1

−h

2

)

μμ

===

ρ

gJ=

ρ

g

v

=

v

dlLLK

ρ

gK

考虑到通用性，将渗流的二项式定律作为达西定律的一般形式，并按坐标系分

方向列出，x

i

方向的表达式如下：

−

∂p

μρ

U

i

+

K

loss

UU

i

=

C

R

1

U

i

+

C

R

2

UU

i

=

2

∂

x

i

K

perm

其中，μ为动力粘度，K

perm

为渗透率，K

loss

为损失系数，C

R1

、C

R2

分别为线性的

和平方的阻力系数。该式就是CFX多孔介质模型计算的基础。

CFX英文帮助中关于多孔介质的内容剪辑如下：

Flow in Porous Media

FlowinporousmediainANSYSCFXcanbecalculatedusingeitheramodelformomentum

loss or a full porous model. The momentum loss model is available in fluid domains, while

the full porous loss model is only available in porous domains.

Darcy Model

The porous model is at once both a generalization of the Navier-Stokes equations and of

Darcy'eusedtomodelflowswhere

the geometry is too complex to resolve with a grid. The model retains both advection and

diffusion terms and can therefore be used for flows in rod or tube bundles where such

effects are important.

Inderivingthecontinuumequations,itisassumedthat‘infinitesimal’controlvolumesand

surfaces are large relative to the interstitial spacing of the porous medium, though small

,givencontrolcellsandcontrolsurfaces

are assumed to contain both solid and fluid regions.

Thevolume porosity

γ

at a point is the ratio of the volume

V'

available to flow in an

infinitesimal control cell surrounding the point, and the physical volume

V

of the cell.

Hence:

V'=γV

(Eqn. 234)

It is assumed that the vector area available to flow,

A'

, through an infinitesimal planar

control surface of vector area

A

is given by:

A'=K⋅A

ij

(Eqn. 235)

where

K

=(

K

)

is a symmetric second rank tensor, called thearea porosity tensor.

Recall that the dot product of a symmetric rank two tensor with a vector is the vector.

K

⋅

A

=

KA

j

ANSYS CFX presently only allows

K

to be isotropic.

The general scalar advection-diffusion equation in a porous medium becomes:

iij

∂

(γρΦ)+∇•(ρK⋅UΦ)–∇•(ΓK⋅∇Φ)=γS

∂t

(Eqn. 236)

In addition to the usual production and dissipation terms, the source term

S

will contain

transfer terms from the fluid to the solid parts of the porous medium.

In particular, the equations for conservation of mass and momentum are:

∂

γρ+∇•(ρK⋅U)=0

∂t

and:

T

∂

(γρU)

+

∇•(ρ(K⋅U)⊗U)

–∇•(

μ

e

K

⋅(

∇U

+

(∇U)

))

∂t

(Eqn. 237)

(Eqn. 238)

=–γR⋅U–γ∇p

(Eqn. 239)

where

U

is the true velocity,

μ

e

is the effective viscosity - either the laminar viscosity or a

turbulent quantity, and

R=(R)

represents a resistance to flow in the porous medium.

This is in general a symmetric positive definite second rank tensor, in order to account for

possible anisotropies in the resistance.

In the limit of large resistance, a large adverse pressure gradient must be set up to balance

the resistance. In that limit, the two terms on the r.h.s. of(Eqn. 239) are both large and of

opposite sign, and the convective and diffusive terms on the l.h.s. are negligible. Hence,

(Eqn. 239) reduces to:

ij

U=–R

–1

⋅∇p

(Eqn. 240)

Hence,inthelimitoflargeresistance,youobtainananisotropicversionofDarcy'slaw,with

permeability proportional to the inverse of the resistance tensor. However, unlike Darcy's

law,youareworkingwiththeactualfluidvelocitycomponents

U

,whicharediscontinuous

at discontinuity in porosity, rather than the continuous averaged superficial velocity,

Q=K⋅U

Heat transfer can be modeled with an equation of similar form:

H

∂

(γρH)

+

∇•(ρK⋅UH)

–∇•(

Γ

e

K⋅∇H)

=

γS

∂t

(Eqn. 241)

where

Γ

e

isaneffectivethermaldiffusivityand

S

the porous medium.

H

containsaheatsourceorsinktoorfrom

Directional Loss Model

From the general momentum equation for a fluid domain:

∂(ρU

i

)

∂

(

ρU

j

U

i

)∂τ

ji

M

∂p

-+ρg

i

+--------+S

i

-----------------

+------------------------=–------

∂t∂x

j

∂x

i

∂x

j

the momentum source,

S

i

can be represented by:

M

(Eqn. 242)

S

i

=–

CU

i

–C

where:

•

•

•

•

Darcy’s Law

MR1R2

UU

i

+S

i

spec

(Eqn. 243)

C

C

R1

R2

is a linear resistance coefficient

is a quadratic resistance coefficient

contains other momentum sources (which may be directional)

S

i

spec

U

and

U

are superficial velocities

A generalized form of Darcy’s law is given by:

ρ

μ

∂p

–-------=

--------------

U

i

+

K

loss

--

UU

i

2K

perm

∂x

i

where:

•

•

•

(Eqn. 244)

μ

is the dynamic viscosity

K

perm

is the permeability

K

loss

is the empirical loss coefficient

ImplementationComparing(Eqn. 243) with(Eqn. 244), the following coefficients are set:

in ANSYS CFX

C

R1

μ

=

---

K

,

C

R2

ρ

=

K

loss

--

2

(Eqn. 245)

Data may sometimes be expressed in terms of the true velocity, whereas ANSYS CFX uses

superficial velocity. If so, the coefficients are represented by:

R1

C

μ

=

------

γK

,

C

R2

K

loss

ρ

=--------------

2

2γ

(Eqn. 246)

where

γ

is the porosity.

Porosity Settings Tab

ThePorositySettingstabiswherethegeneraldescriptionofaporousdomainisspecified

for the simulation.

Area PorosityArea Porosity represents the fraction of physical area that is available for the flow to go

through. The default setting is Isotropic.

VolumePorosityVolume Porosity is the local ratio of the volume of fluid to the total physical volume.

Loss Modelscase,the

loss is specified using either linear and quadratic coefficients, or permeability and loss

coefficients. When specifying the loss coefficients, it is important to properly set theLoss

Velocity Type.

•

•

For details, seeIsotropic Loss Model (p.27 in "ANSYS CFX-Solver Modeling Guide").

For details, seeDirectional Loss Model (p.27 in "ANSYS CFX-Solver Modeling Guide").

Isotropic Loss Model

Isotropic momentum losses can be specified using either linear and quadratic resistance

coefficients, or by using permeability and a loss coefficient. This model is appropriate for

isotropic porous regions.

Permeability and Loss Coefficient

Thismodelspecifiescoefficientsforpermeabilityandloss,inthegeneralizedformofDarcy’s

Law. For details, seeDarcy’s Law (p.67 in "ANSYS CFX-Solver Theory Guide").

Note:The velocity solved by the code (and assumed by the model) is the superficial fluid

velocity. In a porous region, the true fluid velocity of the fluid will be larger because of the

flow volume reduction. Sometimes a loss model is formulated in terms of true velocity

isthecase,thespecifiedcoefficientsmustbeadjusted

accordingly: the permeability must be multiplied by the porosity, and the loss coefficient

must be divided by the porosity squared.

Linear and Quadratic Resistance Coefficients

An isotropic momentum source may also be formulated using linear and quadratic

resistancecoefficients

C

R1

and

C

R2

.Thesecoefficientsmayberelatedtothepermeability

and loss coefficients (mentioned above) as follows:

μ

C

R1

=-------------

K

perm

ρ

C

R2

=

K

loss

--

2

(Eqn. 14)

(Eqn. 15)

Directional Loss Model

For many applications, a certain resistance loss is desired in a specified direction, with flow

thecasewhenyouwishtomodeltheeffectof

flowstraighteningdevicessuchashoneycombs,porousplates,andturningvaneswithout

modeling the details of the flow around the obstacles. For situations like this, ANSYS CFX

allows the independent specification of loss for the streamwise and transverse directions.

For both the streamwise and transverse directions, both types of the loss formulations

available for the isotropic loss model are available. In many cases, however, the loss

coefficientsareknownonlyforthestreamwisedirection,andyouknowonlythattheflowis

inhibited in the transverse direction. When this occurs, you may select theStreamwise

case,thetransversecoefficients

aretakentobethespecifiedfactortimesthestreamwisecoefficients.(Ifthestreamwiseloss

includes a permeability, the implied transverse permeability is divided, not multiplied, by

this factor). The transverse multiplier is typically taken to be about 10-100.

In some cases, you may wish to only inhibit the transverse flow without having any

case,theZeroLossoptionmaybeselectedforthestreamwiseloss.

Ofcourse,ifthisischosen,theStreamwiseCoefficientMultiplierisnotappropriateforthe

transverse loss, because it will result in zero transverse loss.

In all cases, the directional loss model requires the streamwise direction to be specified. It

may be described in either Cartesian or Cylindrical coordinates.