Theory of turbulence and modeling of turbulent transport in the atmosphere

1

Размещено на http://www.allbest.ru/

1. Dynamics of boundary layer

Boundary layer structure

During the last twenty years mathematical modeling of turbulent flows of fluid has been successfully developed in several directions at once [1, 19-54, 59-70, 74-128]. Methods of direct numerical simulation (DNS) [66, 116], large eddy simulation (LES) [140], and different models, based on Navier-Stokes equations averaged according to Reynolds's method [28-38, 44, 51] have to do with these directions. The theory of hydrodynamic instabilities and transition to turbulence was proposed, which is based primary on the mathematical ideas about behavior of the dynamical systems [141-142]. The fractal geometry theory developed by Mandelbrot [143] has been used to explain the chaos and intermittence in the hydrodynamic turbulence [144-145]. To obtain the numerical solutions of applied multidimensional problems the effective numerical algorithms have been created [146-147].

The boundary layer is a typical self organized flow formed around any rigid body moving in the viscose fluid at high Reynolds number. To illustrate the common problems of the boundary layer theory let us consider the structure of the boundary layer on the flat plate in adverse pressure gradient - see figure 6.1. This flow includes the laminar boundary layer (1), the transition flow (2), the turbulent boundary layer (3) and the separated turbulent flow (4).

The laminar boundary layer is a well predicted and sufficiently investigated flow. But this flow is not a stable at high Reynolds number, because it can be like an amplifier for the waves of small amplitude.

The transition layer has a complex structure considered by many authors [62, 141, 145, 149-151]. As it was shown by Jigulev [149] and Betchov [150] this flow domain includes seven sub-regions:

the laminar flow region in which the small disturbances are generated. This part of flow is considered often as a starting point of transition layer. The Reynolds number of initial point of transition layer is a very sensitive to the boundary conditions on the wall and in the outer flow. The estimated value of the Reynolds number of transition is and as high as ;

the quasi-laminar flow region in which the amplitude of linear waves (called the Tollmien-Schlichting waves) grows up to the critical value . The typical scale of this region is about , where is a local thickness of the boundary layer;

the nonlinear critical layer where the interaction between waves and main flow leads to the new unstable state. The typical scale of this region can be estimated as ;

3D waves region with scale . In this region initial two-dimensional waves are transformed into three-dimensional waves;

the region of the secondary instability in which the short length waves are generated. The typical scales of this zone are about , ;

the Emmons sports region with typical scales , . In this part of flow the non-equilibrium process leads to the turbulent spectrum of velocity fluctuations;

the initial region of the turbulent flow in which .

The transition from the laminar flow to the turbulent flow is a very attractive phenomenon from the mathematical point of view. Really the initial laminar flow, which is not consisting of any chaotic waves, then suddenly transforms to the state with a chaotic behavior. This problem of transformation called "dynamical chaos" has been investigated by many authors (see for instance [142, 145]).

The theory of the "dynamical chaos" is based mostly on the analyses of the simplifier dynamical systems (Lorenz-like chaos) which can't be used directly for the boundary layer problem.

The turbulent boundary layer is characterized by chaotic pulsation of the flow parameters. The surface which separates the turbulent stream from the outer flow looks like a rough surface. The thickness of the turbulent boundary layer in zero pressure gradient increases with a distance approximately as a power function, and the skin friction coefficient slowly decreases with the Reynolds number increasing as where (see Schlichting [61]).

Figure 6.1: A) The boundary layer on the flat plate in adverse pressure gradient: 1 - laminar boundary layer; 2 - transition layer; 3 - turbulent boundary layer; 4 - turbulent separated flow; B) the thickness of the laminar boundary layer in the air flow at ; C) the mean height of the separating boundary layer according to Simpson et al [148]

The turbulent boundary layer in adverse pressure gradient separates out from the rigid surface and the boundary layer thickness increases as it is shown in Figure 6.1,c. This part of the boundary layer is not so well predictable as a laminar flow, thus till now the separated turbulent boundary layers were studied only in partial cases primary by experimental way (see Simpson et al [148]).

The turbulent boundary layer can be modelled on the theory of turbulence which was explained in Chapter 2. But it is a very interesting fact that the laminar flow and transition layer also can be described by the equation system (2.14) derived from the Navier-Stokes equations (NSE) due to the special type of transformation (2.1). Let us consider the application of the turbulence theory to the quasi-laminar boundary layer, i.e. to the boundary layer flow which has some symptoms of turbulent flow.

2. Laminar boundary layer

The general solution for the laminar flow can be found on the base of the boundary layer approximation of the Navier-Stokes equations in the Prandtl's form:

(6.1)

Here the pressure gradient is given by equation (4.17), thus

. (6.2)

To derive model (6.1) from the Navier-Stokes equations we should suppose that

the laminar boundary layer is a two-dimensional flow, i.e. ;

the normal to the wall velocity gradient sufficiently exceeds the parallel to the wall velocity gradient, i.e. ;

the normal to the wall pressure gradient is so small that it can be neglected, therefore the pressure distribution is described by the Bernoulli equation (6.2).

It can be shown that the sufficient condition, to satisfy suppositions b)-c), is that the Reynolds number computed on the distance from the plate edge has an extremely high value, i.e. .

Boundary conditions for the quasi-linear diffusion equation (6.1) can be set as follows:

 (6.3)

The first equation (6.1) can be satisfied automatically if we define a flow function as follows

(6.4)

Problem (6.1)-(6.3) has a self-similarity solution for the boundary layer in a zero pressure gradient. In this case , thus the first and third condition (6.3) are identical that means that a solution of this problem depends on the universal variable . Put , then the velocity components can be rewritten as functions of the universal variable, i.e.,

(6.5)

Substituting these expressions in the second equation (6.1) one can find that the universal function is described by the following equation (see, for example, [51, and 58]):

(6.6)

The boundary conditions for equation (6.6) (these conditions can be derived from (6.3)) have a form

(6.7)

The problem (6.6-6.7) can be solved numerically using the algorithm described above in subsection 2.4.2. For the initial iteration one can put (see [51]) that gives in practice the precise solution. Obviously that it's impossible to satisfy last condition (6.7) in a numerical procedure. Hence instead of it as usual the boundary condition in the outer region has used, where [51]. Thus the boundary layer depth can be defined as a point where, for instance, , i.e.

(6.8)

This function is shown in Figure 6.1,b to illustrate the typical scale of laminar boundary layer in the air flow at . Therefore the universal variable can be presented as , where is the boundary layer characteristic thickness

The boundary layer thickness is not a constant; it slowly increases down to the stream so that

(6.9)

This equation gives the normal to the wall velocity scale which can be defined as . With two characteristic scales of velocity equations (6.5) can be rewritten as follows:

(6.10)

The normalised velocity profiles in the laminar boundary layer are shown in Figure 6.2. The normal to the wall velocity normalised on the scale has a limit value at : . The positive value of this velocity component means that the stream lines starting from the boundary layer then penetrate in the outer flow region.

Figure 6.2: The normalised velocity profiles in the laminar boundary layer in zero pressure gradient: 1 - ; 2 -

The normal to the wall velocity scale decreases with distance as . Thus near the transition layer this scale has a very small value which has never been taken into account in the theory of transition to turbulence.

The skin friction coefficient can be defined for the laminar flow as . Substituted in this formula the numerical value of the second derivative, , which was calculated above, we have .

The self-similarity solutions (6.5) found for the laminar flow (called the Blasius flow) is only type of the self-similarity solutions of the Navier-Stokes equations (NSE). Let us give a proof that the Blasius flow can be described by equation system (2.14). Really all solutions of the equation system (2.14) which was derived from NSE are presented by the self-similarity functions. Therefore, we can select from (2.14) also solution for the Blasius flow. First of all note that in this two-dimensional flow , and , hence we have

 (6.11)

,

Here . Let , then the generalised form of (6.5) and (6.6) can be found from first eq. (6.11) and from the definition of immediately as follows

(6.12)

.

The Blasius solution corresponds to the special case when

, . (6.13)

In this case the second eq. (6.12) has a form

(6.14)

The boundary layer approximation (6.1) is applicable only for very high Reynolds number, i.e. for . Hence the term in the brackets which is proportional to can be neglected in (6.14) and finally we have equation (6.6).

3. Transition to turbulence

Continuous transition to turbulence

Passing through the transition layer the laminar stream transforms into the turbulent flow. There are several models of transition to turbulence (see [58, 141, 145, 149] and other). From the point of view of the turbulence theory considered above the parameter characterized the dynamical roughness structure, i.e. , increases in the transition layer from a zero up to , and the second turbulent velocity scale, , increases from a zero up to . Consequently the 2D laminar Blasius flow transforms into 3D turbulent flow.

The general solution (2.16) of the turbulent incompressible flow model (2.14) can be used to analyze the transition from the Blasius flow to the turbulent flow. Put in this solution then the random velocity components can be written as follows

, (6.15)

,

where .

Put as in the case of the Blasius flow then we have

 (6.16)

where a function satisfies to equation

(6.17)

with boundary conditions

. (6.18)

Put in (6.17) as for the Blasius flow solution, therefore

, (6.19)

where is an arbitrary function.

3D Transition to turbulence

The first scenario of spatial continuous transition to turbulence is that and . In this case the boundary conditions (6.18) are similar to the Blasius flow conditions. For we have exactly the Blasius flow solution - see Figure 6.3. Put then the dynamical roughness parameters are given by

, . (6.20)

As it follows from this equations if increases then the dynamical roughness parameters also increase and the laminar boundary layer velocity profile (the Blasius profile (1) in Figure 6.3) transforms into the turbulent boundary layer velocity profile (6) - see Figure 6.3.

Figure 6.3: Continuous transition from the laminar flow (the Blasius velocity profile (1)) to the turbulent flow (the logarithmic velocity profile (6)). Profiles 1-6 are computed on (6.17)-(6.18) for and for respectively

Figure 6.4: Continuos transition to turbulence: 1 - the mean velocity profile in the turbulent boundary layer according to Van Driest [65], 2, 3 - the mean velocity profiles in the transition layer computed on the model (6.24), (6.25) for respectively

Theoretically the logarithmic profile in this model can be only at , but practically the logarithmic asymptotic is realised for - see Figure 6.4. It can be explained by the asymptotic behavior of a solution of equation (6.17) at : , where are some parameters. Obviously, that for the Blasius flow , and in a common case for . Therefore can be estimated for as follows

(6.21)

.

Substituted this expression in the first equation (6.15) and supposed that one can derive the asymptotic formula for the streamwise velocity gradient, i.e.

(6.22)

Here for . Used the inner layer variables for the mean velocity scaling the last equation can be rewritten as follows

. (6.23)

Calculated the exponent for we have . Thus in this case the power function factor in the right part of equation (6.23) is about unit for hence equation (6.23) leads to the logarithmic profile asymptotic

Here is the Karman constant. Using the relationship third boundary condition (6.18) in a case of mean velocity profile can be transformed as follows

Using the inner layer variables we can rewrite the model of spatial transition to turbulence in the form

, (6.24)

, ,

, ,

where , , (as for the Blasius flow), , . The boundary conditions for this model are given by

(6.25)

The mean velocity profiles computed on the model (6.24) for (the solid lines 2,3) together with the mean velocity profile in the turbulent boundary layer proposed by Van Driest [65] (the symbolised line 1) are shown in Figure 6.4. The logarithmic asymptotic of the profile (3) has a form , where .



Figure 6.5: Continues transition to turbulence: 1 - the mean velocity profile in the turbulent boundary layer according to Van Driest [65], 2-4 - the mean velocity profiles in the transition layer computed on the model (6.24), (6.26) for and respectively

The main difficulty of model (6.24) is that the estimated streamwise velocity profile (3) is not really the logarithmic profile in the turbulent boundary layer over smooth surface as it should be, but it is the logarithmic profile which can be in the turbulent boundary layer over a rough surface. Thus the skin friction coefficient of this flow is higher then in the turbulent boundary layer with the similar thickness and free stream velocity.

The second scenario of continuous transition to turbulence is that and . In this case the transition layer model is identical to (6.24) with boundary conditions

(6.26)

Using the constant of the logarithmic profile, we can estimate an additional parameter, i.e. . The mean velocity profiles computed on the model (6.24), (6,26) for and are shown in Figure 6.5 - the solid lines (2-4) respectively.

This model consists of three parameters while in the turbulent boundary layer there is only one parameter - the Karman constant.

2D Transition to turbulence

It's a well known fact that the transition layer includes the quasi-laminar flow region in which the amplitude of linear Tollmien-Schlichting waves grows up to the critical value . This 2D transition zone can be described by the model (6.24) rewritten in the outer region variables. Put in system (6.24) . In this case we have 2D transition model

, , (6.27)

with boundary conditions

(6.28)

where (as for the Blasius flow), .

This model depends on only free parameter . The streamwise velocity profiles computed on this model for are shown in Figure 6.6 left, by the solid lines (1-3) respectively. This solutions are similar to the Blasius flow solutions for a laminar flow with gas injection (see Cebeci & Bradshaw [51]).

The numerical data for dimensionless velocity gradient, , are plotted in Figure 6.6 right. These data can be approximated as follows (see the solid line in Figure 6.6 right)

(6.29)

where as for the Blasius flow.

There is no a logarithmic profile in 2D flow, but in this type of transition the drag increases up to the value which is typical for the turbulent boundary layers. Really, substituting an expression of the roughness surface parameter in formula (6.29) we can derive an equation for the skin friction coefficient as follows

(6.30)

where is the skin friction coefficient for the Blasius flow.

Supposing in the equation (6.30) that and expressing the skin friction coefficient in an explicit form, finally we have

(6.31)

For the last equation exactly gives the Schlichting formula for the skin friction coefficient in the turbulent boundary layer, i.e. (see Schlichting [61]). It means that the transition to turbulence is characterized mostly by the parameter which has a high value in the beginning of the turbulent boundary layer. Hence in this case another scaling should be proposed to balance the dynamical roughness parameters effect on the turbulent flow.

Figure 6.6: The velocity profiles in 2D transition layer calculated on (6.27)-(6.28) for - the solid lines (1-3) respectively (left); the normalised skin friction coefficient versus the dynamical roughness surface parameter (square symbols) together with the approximated solid line (right)

On the other hand as it was shown in subsection 2.4.3, in the turbulent boundary layer also we have . Thus the typical value of the dynamical roughness gradient parameter is about . Taken into account that we can conclude that the transversal gradient , therefore . Obviously that a negative value also available with the same probability as a positive value, because as it follows from second equation (6.15) , and hence the mean transversal velocity .

4. Spectral characteristics of turbulent flows

turbulence laminar flows

The turbulent boundary layer can also be described by equations system (6.24). Put in the first equation (6.24). Substituting an universal variable and an universal function in the second and third equations (6.24) we have

, , (6.32)

,

Here is the Reynolds number calculated on the dynamical roughness parameters. The boundary conditions for this model can be proposed as follows

(6.33)

Where is a free parameter. To establish this parameter it can be claimed that the streamwise mean velocity profile has a logarithmic asymptotic at . Surmising that we have from the first equation (6.32) , and therefore .

The last equation gives the continuous spectrum of the turbulent scales . On the other hand the mean velocity profile in the logarithmic layer can be characterized by one scale. To solve this problem note, that equation can be rewritten in the form

, (6.34)

Here is the second turbulent velocity scale.

For an uniqueness of the mean velocity profile one can suppose that for the second turbulent velocity scale has a stable value at small variations of the parameter , i.e. (or for the Karman constant has a stable value, i.e. ). It gives and for . The fundamental turbulent boundary layer scale can be defined as . The mean velocity profile calculated on this model for , and for is shown in Figure 2.4, a.

The function can be considered as a spectral density. The inverse length scale versus the Reynolds number is shown in Figure 6.7, a. This type of a spectral density is similar to the spectral density of the streamwise velocity fluctuations in the turbulent boundary layers. The function is shown in Figure 6.7, b. This type of spectrum is similar to the spectral density of the transversal velocity pulsation (see Tennekes & Lumley [152]). Both functions represent the constructive model of the hydrodynamic chaos in this theory of turbulence.

To compare the spectral density with experimental data we can suppose that , where is the characteristic radian frequency. Therefore the Reynolds number calculated on the dynamical roughness parameters depends on the frequency as

(6.35)

Here is the typical turbulent length scale of the streamwise velocity pulsation, is the flow wave number

(6.36)

There are several models which have been proposed to describe the spectral density in the turbulent boundary layers (see Simpson et al [148], Tennekes & Lumley [152], Perry et al [154] and other). The widely used spectral density in the logarithmic layer is given by . Instead of this we suggest that the spectral characteristic of the turbulent flow is related to the eigen spectrum of the problem (6.32)-(6.33), thus

(6.37)

Here is the normalizing factor which can be calculated from (6.36).

Figure 6.7: The inverse length scale (a) and the normalised velocity scale (b) versus the Reynolds number calculated on the dynamical roughness parameter

Suggesting that the flow wave number in the logarithmic layer depends on the dynamical roughness Reynolds number as the linear function, i.e. , we have

, (6.38)

therefore .

The spectral density computed on (6.35), (6.37) is shown in Figure 6.8 (solid lines) together with the experimental data by Hussain & Reynolds [153] obtained in the turbulent channel flow. The experimental values and estimated parameters for the data are listed in Table 6.1. The boundary layer height, the mean velocity on the channel axis , the Reynolds number of the turbulent boundary layer , and .

As it was established both spectral density parameters and slowly depend on the distance from the wall. In the inner layer the experimental data is in a good agreement with the predicted spectral density (a, b). But in the outer region the correlation is not so good (c, d). It can be explained by the mixed layer contribution in the velocity pulsation.



Figure 6.8: The spectral density of the streamwise velocity pulsation in the turbulent channel flow computed on (6.37) (solid lines) and the experimental data by Hussain & Reynolds [153] measured at the distance from the wall (a), (b), (c), and

Table 6.1 Input data for Figure 6.8

Figure	6.8 a	6.8 b	6.8 c	6.8 d
	4.9	11.7	106	770
	6.76	5.45	4.61	4.61
	1.01	1.82	2.27	1.36

The local rate of dissipation of the mean flow kinetic energy in the logarithmic layer is given by

The optimal parameter brings a maximum for the second turbulent velocity scale and a minimum for the Karman constant. In turn the minimum of the Karman constant is related to the maximum of the local rate of dissipation of the mean flow kinetic energy.

Using the dynamic roughness parameter we can propose the scaling for the local rate of dissipation of the mean flow kinetic energy in the logarithmic layer as follows

,

Here is the scale of time in the inner layer.

Apparently it means that the Karman constant should be determined as . Therefore two another constants of the theory are given by , . This is the final closure. Hence this theory of turbulence is the completely closed theory, because all parameters have been calculated within the theory from the "first principles".

Размещено на Allbest.ru

...