Durst F. Fluid Mechanics: An Introduction to the Theory of Fluid Flows

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18.7 Turbulence Models 573

18.7.4 One-Equation Eddy Viscosity Models

The class of turbulence models, discussed in Sect. 18.7.3, tries to describe the

momentum transport properties of turbulent ﬂows with the help of a single

parameter, namely the Prandtl mixing length. The latter is introduced into

the considerations in a geometry-speciﬁc way and is stated in the form of

an algebraic equation, where the velocity characteristic of the turbulence is

introduced via the mixing length and the gradient of the mean velocity ﬁeld.

In order to achieve a model expansion, the characteristic velocity typical of

the turbulent momentum transport is set as follows:

= k

with k =



. (18.207)

With the characteristic length l

of one-equation models, thus ν

=(l

)

results. Here k is the local turbulent kinetic energy, which is described by the

following transport equation:

∂k

∂x

∂

∂x

−



+2ν



∂u



∂x

−



−



∂U

∂x

−2ν



∂u



∂x



(18.208)

so that the turbulent transport properties in (18.208) change with the modi-

ﬁcations of turbulence in a ﬂow ﬁeld. It is customary in turbulence modeling

to rewrite the above equation as follows:

∂k

∂x

diﬀusion

  

−

∂D

∂x

production



P −

dissipation



, (18.209)

where the following relationships hold:

= −

−



+2ν



∂u



∂x

−



(18.210)

and

P = −



∂U

∂x

;  = −ν

∂u



∂x

∂u



∂x

. (18.211)

With the modeling assumptions for D

, P and , customary in present-day

turbulence modeling research, the following modeled k equation results:

∂k

∂x

∂

∂x



ν +



∂k

∂x





 

diﬀusion

+ ν



∂U

∂x

∂U

∂x



∂U

∂x

  

production

− C

)

  

dissipation

(18.212)

574 18 Turbulent Flows

The second term on the right-hand side of (18.212) represents the production

of turbulence by the mean ﬂow ﬁeld, which was stated as follows:

−



∂U

∂x

= ν



∂U

∂x

∂U

∂x



∂U

∂x

. (18.213)

The ﬁrst term on the right-hand side of (18.212) also contains σ

,aquantity

which indicates how the turbulent diﬀusion of k is related to the turbulent

momentum diﬀusion ν

, i.e. σ

states the relationship of turbulent momen-

tum dissipation to energy dissipation. The third term on the right-hand side

of (18.212) states the turbulent dissipation. For this term, order of magnitude

considerations suggest, for equilibrium ﬂows:

P ∼

∼  ;  = C

)

. (18.214)

The constants introduced into this one-equation turbulence model have to be

determined from experiments. Here, one makes use of the two-dimensional

form of the k equation for a boundary-layer ﬂow:

∂k

∂x

∂k

∂y

∂

∂y



ν +



∂k

∂y



+ ν



∂U

∂y



−c

)

. (18.215)

In the turbulent equilibrium range of the ﬂow (inertial sub-range), it holds

that P = , i.e. it can be stated that:





=+C

)

. (18.216)

On taking into account that the following relationships are valid:

= −ρν





≈−τ

(18.217)

and considering that τ

/ρ = u

, one obtains τ

/ρ



∂U

∂y



= c

3/2

)

,and

with

∂U

∂y

≈

1/2

)

it can be stated that

= C

−

. (18.218)

As in Sect. 18.7.3, for the one-equation k-l model, assumptions were chosen

in turbulence considerations which “in wall boundary layers” take into ac-

count the closeness of the wall and which lead to diﬀering assumptions for

)

values, depending on whether one considers the diﬀusion term or the

dissipation term:

18.7 Turbulence Models 575

)

1,ν

= C

κy [1 − exp (−A

)] with A

=0.016 (18.219)

and

)

1,D

= C

κy [1 − exp (−A

)] with A

=0.26, (18.220)

where R

represents the Reynolds number formed by wall

disturbance:

)

= l

. (18.221)

From boundary layer data, the value C

≈ 0.09 results. With this value,

it is now possible to integrate the Reynolds equations employing the k-l

one-equation turbulence model.

In order to determine the C

value, the equilibrium region of a boundary

layer was employed. Figure 18.18 shows where this region is located.

It is also customary to employ other characteristic ranges of turbulent

ﬂows to determine the “free constants” in turbulence models. Altogether the

following ranges are available, introduced here by terms customary in the

English literature on turbulence.

Equilibrium range:

0=P − . (18.222)

Decay range:

∂k

∂x

= −. (18.223)

Wall near

region

Core region

Equilibrium

region

Fig. 18.18 Ranges of turbulent channel ﬂow

576 18 Turbulent Flows

Rapid distortion range:

∂k

∂x

= P. (18.224)

The above equations are useful in discussions of diﬀerent turbulent ﬂow

regions.

18.7.5 Two-Equation Eddy Viscosity Models

Practical experience with turbulence models shows that, as far as their

general applicability is concerned, the k-l one-equation model represents a

considerable improvement over the zero-equation models. However, very good

computations of turbulent ﬂows can only be carried out by zero- and one-

equation models when small ﬂow accelerations or decelerations occur, i.e.

ﬂows with strong pressure gradients can only be computed in an unsatisfac-

tory way with k-l one-equation models. Experience shows that the limitations

of the applicability of k-l one-equation models are due to the algebraic form

of the characteristic length scale l, which often means that the applicabil-

ity remains restricted to such ﬂows which were applied for deriving the k-l

one-equation models. This is the reason for the introduction of two-equation

eddy viscosity models. One of these models is the k- model, which is based

on the solution of the following two diﬀerential equations:

k equation:

∂k

∂t

∂k

∂x

  

= −u

∂U

∂x

  

−

∂

∂x







 

−ν

∂u

∂x

∂u

∂x

  



+ν

∂

∂x

  

(18.225)

where P = production term, T = transport term and  = dissipation term

and D = diﬀusion term.

 equation:

∂

∂t

∂

∂x

= −2ν

∂u

∂x

∂u

∂x

∂U

∂x

  



− 2ν

∂u

∂x

∂u

∂x

∂U

∂x

  



−2νu

∂u

∂x

∂

∂x

δx

  



− 2ν

∂u

∂x

∂u

∂x

∂u

∂x

  



−

∂

∂x

νu

∂u

∂x

∂u

∂x

  



−

∂

∂x

2ν

∂u

∂x

∂p

∂x

  



18.7 Turbulence Models 577

−2ν

∂

∂x

δx

∂

∂x

δx

  

γ>0

+ ν

∂



∂x

δx

  



. (18.226)

The eddy viscosity can be deﬁned with k and , i.e. with u

= k

1/2

and

3/2



, as follows:

= C



Here, k and  are determined from the above-modeled diﬀerential equations.

In this context, it is useful to know that in turbulence modeling the above-

cited equation for k is considered to be suﬃciently well modeled as concerns

practical computations, whereas similarly satisfactory modeling assumptions

do not exist for the  equation. The model equations often used in present-

day ﬂow computations are the following:

k equation:

∂k

∂t

∂k

∂x

= ν



∂

∂x

∂

∂x



∂

∂x

∂

∂x

∂k

∂x

−  + ν

∂

∂x

(18.227)

With ν

∼

0.09



 equation:

∂

∂t

∂

∂x

= c

1





∂

∂x

∂

∂x



∂

∂x

− c

2





∂

∂x



∂

∂x

+ ν

∂



∂x

. (18.228)

Here, the modeling of the last term in the  equation is based on the

assumption:

P ∼ k

 ; ν







. (18.229)

As concerns the boundary layer formulation of the Reynolds and k- turbu-

lence model equations, valid for high Reynolds numbers, they can be given

as follows:

∂U

∂x

∂V

∂y

=0, (18.230)

∂U

∂x

+ V

∂U

∂y

= F

−

∂P

∂x

∂

∂y



(ν + ν

)

∂U

∂y



, (18.231)

∂K

∂x

+ V

∂K

∂y

∂

∂y



∂K

∂y



+ ν



∂U

∂y



− , (18.232)

578 18 Turbulent Flows

∂

∂x

+ V

∂

∂y

∂

∂y





∂

∂y



+ C

1





∂U

∂y



− C

2



, (18.233)

= C



where C

=0.09, σ

=1.0, σ



=1.3, C

1

=1.45, C

2

=2.0.

In order to determine c

,theansatzν



∂U

∂y



=  ; c

= c





0.09 again holds. This can also be determined from measurements of wall

boundary layers.

18.8 Turbulent Wall Boundary Layers

Turbulent boundary-layer ﬂows, whose essential properties are determined by

the presence of a wall, are called wall boundary layers. As classic examples

one can cite

• Internal ﬂows: plane channel ﬂows and pipe ﬂows

• External ﬂows: plane plate ﬂows and ﬁlm ﬂows

These ﬂows are sketched in Fig. 18.19. Their essential feature is the momen-

tum loss to a wall which is characteristic of all ﬂows in Fig. 18.19, i.e. wall

momentum loss exists in all the indicated ﬂow cases and, in addition, the

properties of the ﬂuid, the density ρ and the dynamic viscosity µ characterize

the ﬂuid.

In order to discuss the properties of turbulent boundary layers in an

introductory way, the fully developed, two-dimensional, plane, turbulent

Pipe flows

Film flows

Plane chamber flows

Wall boundar

ers

Normalisation of the flow

data yields

and

All these flows are

characterised by

momentum losses to walls

Fig. 18.19 Examples of internal and external wall boundary layers

18.8 Turbulent Wall Boundary Layers 579

channel ﬂow is subjected to more detailed considerations below. From the

Reynolds equations (see Sect. 18.5.2), the following reduced set of equations

can be deduced for fully developed, plane channel ﬂows with the above-cited

properties:

− momentum equation: 0 = −

∂P

∂x



− ρu





. (18.234)

− momentum equation: 0 = −

∂P

∂x

− ρu



. (18.235)

− momentum equation: 0 = −







. (18.236)

The last partial diﬀerential equation (18.236) can be integrated and yields,

because of the wall boundary condition



= 0, that the correlation u



in the entire plane perpendicular to the plates of the channel has the value



=0.

The integration of the second diﬀerential equation (18.235) yields

P (x

)=P

) − ρu



, (18.237)

where ρ



= f (x

), since in the x

-direction the ﬂow was assumed to be

fully developed. Nevertheless, the above relationship expresses that the pres-

sure in a turbulent channel ﬂow changes slightly over the cross-section. The

change proves to be so small, however, that it can be neglected for practical

considerations of the properties of fully developed, two-dimensional, plane,

turbulent channel ﬂows. Thus, from (18.237) the following results for the

pressure gradient:

∂P

∂x

≈

. (18.238)

Inserted in (18.234), one obtains:



− ρu







 

ges

dτ

ges

. (18.239)

If one introduces, for scaling the above equations, the velocity and length

scales:

and !

(18.240)

the momentum equation (18.239) can be written in general form as follows:

=1−







. (18.241)

580 18 Turbulent Flows

Fig. 18.20 Fully developed turbulent, plane channel ﬂow

Fig. 18.21 Terms of the momentum equation for two-dimensional channel ﬂow

Here the wall coordinate y is introduced: y = H − x

; y

; Re

and









. (18.242)

With these standardized quantities introduced into (18.242), (18.241) can be

derived. The latter relation comprises four terms which can all be seen in

Fig. 18.21.

In Fig. 18.21, the horizontal line represents the value 1. The quantity

−y

/R e

and the quantity −u



in the equation and also dU

/ dy

are

also shown in Fig. 18.21.

From (18.239), we obtain

ges

(h − y)

so that the diﬀerent terms stated in (18.241) can be shown as in Fig. 18.21.

It is evident that the term dU

/dy

represents, over wide ranges of the ﬂow,

the smallest value in the standardized momentum equation (18.241).

18.8 Turbulent Wall Boundary Layers 581

In order to obtain information on dU

/dy

for plane channel ﬂows, laser

Doppler and hot wire measurements were carried at the Institute for Fluid

Mechanics (LSTM) of Friedrich-Alexander-University, Erlangen-N¨urnberg to

achieve U

(y) distributions experimentally. In connection with detailed shear

stress measurements, the presentation of the normalized measured values of

the velocity gradient was achieved in the form





= f

ln y

(18.243)

and from this it was determined that, for high Reynolds numbers, the mea-

sured and normalized mean velocity gradients plotted in Fig. 18.21 can be

described as follows:





= −ln y

+1≡ ln

. (18.244)

From this, U

= e ln y

+ B can be obtained, i.e. the standardized velocity

distribution in a plane channel ﬂow can, over a wide range of the channel

cross-section, be described by a logarithmic velocity distribution:

ln y

+ B with

κ =1/e

B =10/e

. (18.245)

These values were found through the experimental investigations at LSTM

Erlangen.

Figure 18.22 shows that the double-logarithmic plotting yields a line with

gradient −1. This is due to the fact that the logarithmic law is valid for the

normalized velocity distribution.

In the literature there has been a number of investigations to determine

the value of κ and the additive constant B to represent with these the loga-

rithmic boundary velocity law. The following represents a summary of these

investigations and the resultant values. The large variation in the values is

mainly due to the use of measurement techniques which do not permit suﬃ-

ciently local measurements of the mean ﬂow velocities. Furthermore, eﬀects

which arise from ﬂows of low Reynolds numbers were included in the eval-

uations of κ and B in some literature data. If one considers all the possible

inﬂuences, i.e. permits only reliable hot wire and laser Doppler measurements

to enter the evaluations, then one obtains the values indicated in (18.245) for

κ and B (Fig. 18.23).

If one employes only reliable measurements for dU

/dy

, the relationship

stated in (18.241) allows the determination of (



)

values for turbulent

channel ﬂows. Distributions of these turbulent transport terms, plotted in a

normalized form, are shown in Fig. 18.24.

By means of a plane channel measuring test section with glass side

walls and the use of an LDA velocity-measuring system, information on the

582 18 Turbulent Flows

Fig. 18.22 Representations of experimental investigations for determining the

logarithmic wall law

Fig. 18.23 Scatter of the κ and B values in the experimental determination of

= f (y

) for wall boundary layers

turbulent velocity ﬂuctuations existing in ﬂow direction, could also be ob-

tained at LSTM Erlangen. This information is shown in a summarized way

in Fig. 18.25. In this ﬁgure they are compared with corresponding results of

numerical ﬂow computations.

Detailed measurements carried out at LSTM Erlangen have conﬁrmed in-

teresting new results, e.g. that the local turbulence intensity





does

not adopt a constant wall value. The result shows that the constant wall

value of this velocity ratio depends on the Reynolds number of the ﬂow (see

Fig. 18.26). Although small discrepancies can be seen as compared with the