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Theory of turbulence and simulation of turbulent transport in the atmosphere. Part 4

Alexander Trunev Ph.D.

Director, A&E Trounev IT Consulting, Toronto, Canada

Summary

The completely closed model of wall turbulence was derived directly from the Navier-Stokes equation. The fundamental constants of wall turbulence including the Karman constant have been calculated within a theory. This model has been developed also for the accelerated and non-isothermal turbulent boundary layer flows over rough surface

Keywords: accelerated flow, aerosol turbulent transport, atmospheric stratified flow, atmospheric turbulence, atmospheric surfase layer, boundary layer, rough surface, turbulent transport Theory of turbulent flows in temperature and pressure gradients

Аннотация

Теория турбулентности и моделирование турбулентного переноса в атмосфере. Часть 4

Трунев Александр Петрович

к. ф.-м. н., Ph.D.

Директор, A&E Trounev IT Consulting, Торонто, Канада

В работе представлена полностью замкнутая модель турбулентного пограничного слоя, полученная из уравнения Навье-Стокса. Фундаментальные константы пристенной турбулентности, включая постоянную Кармана, определены из теории. Эта модель была развита для ускоренного и неизотермического пограничного слоя над шероховатой поверхностью

Ключевые слова: атмосферная турбулентность, турбулентный перенос, ускоренные течения, пограничный слой, шероховатая поверхность, приземный слой атмосферы, турбулентный перенос аэрозолей

The problem of turbulent flow in the gravitational field with a prescribed gradient of temperature (problem of stationary atmospheric turbulent convection) has been considered. Generalisation of Monin-Obukhov's theory of similarity [21-22] has been given. turbulence boundary laye accelerated

The model of the turbulent boundary layer in pressure gradient has been presented. An application of this model to atmospheric boundary layer, on a scale of which the planet rotation plays an important role has been adopted.

Thermal boundary layer

The problem of mass and heat-transfer in non-isothermal turbulent boundary layers has been studied by many authors [1, 3-12, 15-115]. In the last three decades numerical methods of calculations of heat-transfer have been successfully developed [51, 116].

Let us consider a model of non-isothermal turbulent boundary layer based on the theory of turbulence considered above. In a case of a steady flow third equation (2.10) can be written as

(1)

where ,

is the turbulent scale of temperature, is the heat flux from the rigid surface to the air, is the specific heat at constant pressure of the gas.

The boundary conditions for equation (1) on a smooth wall and for the long distance from the wall are given by

(2)

,

where is a constant which is approximately equal to the Karman constant.

Neglected by the term in the left part of (1) in special case when one can derive a general solution of the boundary value problem (1)-(2) in the form:

(3)

The temperature profile (3) well demonstrates that the main turbulent scale of the thermal layer is not the same as the turbulent length scale of the dynamic boundary layer:

.

Hence in a common case two independent length scales should be defined in the turbulent non-isothermal boundary layer problem.

For the long distance from the wall, , the mean temperature profile (3) is resulted to the logarithmic profile:

where .

If the Prandtl number is about unit, , then the temperature gradient near to the wall essentially depends on the dynamic boundary layer parameters. For this case the temperature gradient in a steady turbulent boundary layer at a given heat flux on the wall can be described by equation (1) rewritten in the form:

(4)

where , ,

is the surface of the thermal sublayer chosen for the turbulence modelling,

.

In this problem the universal parameters can be defined as follows

,

For atmospheric flows when we can suggest that

, therefore .

Used this nomenclatures and boundary conditions (2) one can derived the temperature gradient as a solution of equation (4) as follows

(5)

where , , .

Since the problem (5) with (2) is analogue to that which has been used for the mean velocity profile calculation, thus one can establish the equation connected the constants of turbulence theory which is similar to (2.24), as follows:

(6)

where is the Peclet number calculated on the thermal sublayer surface parameters,

, .

It should be noted that

and that for equation (6) is resulted to (2.24). Thus to define the thermal length scale and parameter with the given , the equation (6) can be used in the same way as the Karman constant has been calculated: is the constant at small variations of the Peclet number if , therefore some critical value can be calculated as the root of this equation.

The value computed on this model practically is a constant, for the range - see Figure 1. The computed stable mean value of the Peclet number is given by

for the Prandtl number in the range . The thermal layer length is defined as

.

The mean temperature profiles calculated on (5) for various Prandtl number can be compared with the model of Cebeci [51] - see Figure 2 (right). It's well known that the Cebeci model based on the turbulent Prandtl number conception and it can be written as follows (see Cebeci & Bradshaw [51]) :

,

The mean velocity gradient in the right part of the second equation is given by Van Driest model (2.29), is the turbulent Prandtl number, is the damping length of the turbulent thermal layer, defined as

,

Here .

Figure 1 The ratio versus the molecular Prandtl number



Figure 2 Mean velocity (left) and temperature (right) profiles in the turbulent boundary layer computed on the Cebeci model (symbols) and on (5) - solid lines

As it is shown in Figure 2 the agreement between the predicted profiles and the widely used models of van Driest and Cebeci in general is good. But the present model depends on only one constant, while, for instance, the Cebeci model depends on 8 constants as minimum.

The thermal layer stays closer to a wall than the viscous sublayer for and farther from it for as it follows from the thermal layer length scale,

.

For the small Prandtl number, , the thermal sublayer is so thick that completely consists of the viscous sublayer. In the case when , the thermal layer practically is independent on the dynamic transition layer, and the mean temperature profile is given by equation (3) - see Figure 3 (left), with thermal length scale

.

In another case when (practically when ) the thermal sublayer is so thin that it is included in the viscous sublayer as a whole. Figure 2.2 shows that in this case the solutions with the negative quantity of the dynamic roughness parameter effects on the thermal layer near the wall. Suggesting that

,

and using the expression

the mean temperature gradient (5) can be rewritten as follows

(7)

where , .

Integrating by parts and using the boundary conditions on the wall: , the last integrals can be transformed as follows:

(8)

The third equation (2.16) can be written in the form

Using this equation and secondary integrating by parts the integrals in the right side of (8) we have:



If , then , therefore we can estimate the contributions of all terms versus the small parameter, , as follows:

The last expression is proportional to for , and to for . Therefore the first integral (8) can be estimated as

The function in the right part of the last expression is proportional to , because in the wall region the normal velocity is about

.

Hence as it follows from above, for the large quantity of the Prandtl number the mean temperature profile depends on the normal velocity as a weak function which is proportional to for and to for .

Using this result it is possible to derive a simple equation for the mean temperature gradient, which is satisfactory agreed with experimental data in a wide range of the Prandtl number. Integrating by parts the term in the right part of second equation (8), we have



where

.

This process can be prolonged and the formal series in powers of can be found. The first and second terms of this series are equal to zero on the wall and for . Hence, the equation, connecting constants of the theory, similar to (5), has in this case a very simple form:

,

that coincides with the thermal layer length scale specified above for .

Substituting the derived expressions in equation (7) and using the first term of the series one can write the mean temperature gradient as

(9)

where

is given by (2.19).

The equation (9) contains the free parameter

,

which is the Peclet number. As it was established in the numerical experiments, to agree (9), for example, with the model of Cebeci [51], it is enough to consider this parameter as a function, which depends on the Prandtl number. The numerically approximated function, , is given by

in the range of the Prandtl number from 2 up to 103. Hence, the Peclet number is the negative quantity. In turn it means that . Therefore the mean temperature profile depends mainly on the dynamic roughness surface disturbances, which move to the wall for . The mean temperature profiles computed on equation (9) are shown in Figure 3 (right) by solid lines together with the profiles computed on the model of Sebeci [51] - the symbolized lines. The agreement between two models in general is good.

Figure 3 Mean temperature profiles in the turbulent boundary layer computed for the small (left) and large Prandtl numbers (right) on the model of Sebeci [51] - symbols, and on the proposed model - solid lines

The mean temperature profiles computed on the model (9) are well correlated with the model of Sebeci also for , because the parameter

is the small quantity even for .

This theory can be formulated shortly, using the transformation formula for in (7) as follows

Then substituted the approximation of the normal velocity near the wall,

,

in the right part of the last expression, finally we have

Obviously that this approximation is relatively good for as well as for . Since the equation (9) is based on this approximation, it can be established that the normal velocity contribution in the mean temperature profile is a relatively small value (this contribution depends on the second derivation of the normal velocity on the wall which is estimated for the mean flow as

)

From the data shown in Figure 2.8 one can conclude that the turbulence intensity normal velocity profile has a maximum for where the mean temperature profile - see Figure 3 (right), is described by a logarithmic function. Thus as it follows from equation (9) the interaction of the temperature and velocity fluctuations for the large Prandtl number primarily depends on the function which is proportional to the dynamic roughness parameters, i.e.,

.

In the mixed layer the common expression of the mean temperature profile is given by

(10)

where ,

is the mean temperature profile in the inner layer.

As it has been established the mean temperature profile in the outer region of the turbulent boundary layer depends on two parameters: and . But the thermal layer scale in the outer region is not equal to the dynamic layer scale .

Apparently, that model (9) seems to be more simplified then (7). Nevertheless the model (7) is preferable for the theoretical consideration because using this model it is possible to calculate the Peclet number as function of the Prandtl number, , without any experimental data.

Turbulent flow in the stratified surface layer

Usually the atmospheric boundary layer has a thermal stratification in the bottom part due to the solar radiation and the heat-transfer from the ground surface to the air. In common case the temperature stratification produces some dynamic effect in the air flow such the local free convection due to buoyancy forces. The consideration of this problem based on the similarity theory was given by Monin & Obuchov [21]. The main parameter of the Monin & Obuchov theory is the length scale

where is the air temperature in the surface layer, is the heat flux from the ground to the air.

As it was established (see [21-22]) for the stratified layer is stable, at is unstable, for neutral stratification is attained what means that there is no thermal effect on the turbulent flow. Let us consider the Monin & Obuchov theory based on the present theory of turbulence.

The introduced parameter B in the right part of the second equation (2.12) is connected with the length scale L introduced by Monin & Obuchov [21-22] as follows

Therefore, the stable flows are realized for , the unstable flows - for , and the neutral stratified flows for . Thus to estimate the turbulent stratified flow parameters based on the present theory of turbulence we should establish the boundary conditions and the theoretical constants (or the parameters with stars). The proposed model can be written as

, (11)

,

where , ;

, , .

As it has been established the buoyancy force parameter depends on the type of stratification as a discontinuous function

This dependence can be explained by the influence of the large scale random amplitudes with (a maximal root of the equation)

for

on the mean velocity profile in a case of the stable stratification. It can be shown that the buoyancy force term in the right part of the second equation (2.12) increases as third degree of the ratio of two scales, i.e. as .

The boundary condition for equation system (11) on the imaginary smooth surface is given by

(11,a)

In the outer region in a case of unstable stratification the boundary conditions can be established for as follows

, (11, b)

where parameters , can be deduced from the experimental data.

In a case of a stable stratification the boundary conditions can be formulated at the level

, (11, c)

where the given values, also can be defined by the empirical way.

The mean velocity and temperature profiles in the stable and unstable turbulent flows, computed on the present model (11), are shown in Figures 4 and 5 (solid lines 2). For a comparison in these Figures the profiles computed on the Monin-Obuchov model [21-22] are shown by the broken lines 1. It is a well known fact ( see, for instance, [23-24, 26-27, 117-119]) that this model can be formulated for the mean velocity and temperature gradient as follows

(12)

The momentum and heat universal functions have been established several times from experimental data by Businger et al [23], Dyer [26], Van Ulden & Holtslag [27], Hogstrom [117], Beljaars & Holtslag [118] and other.

For the Karman constant the universal functions have been estimated by Pugliese et al [119] in the form

(13)

To agree the mean velocity and temperature profiles computed on the models (11) and (12) for the model (12) has been regularised at as follows

(14)

The boundary conditions on the imaginary smooth wall for this equation system can be given as

, (14')

The boundary conditions are chosen here in the form (14') since the model (12) with universal functions (13) describes actually the turbulent flow over a rough surface, and the model (11) has been formulated for the turbulent flow over a smooth surface. To derive these conditions, note that the mean velocity shift in the neutral stratified turbulent flow over rough surface given by equation (3.2) can also be used for the stratified flows.



Figure 4 The mean velocity profiles in the stratified surface layer computed on (11) - the solid lines, and (14) - the broken lines

Figure 5 The mean temperature profiles in the unstable (left) and stable (right) turbulent boundary layers computed on (11) - the solid lines, and (14) - the broken lines

As it follows from the data shown in Figures 4-5 in a case of an unstable stratification (L=-112.43 m, L=-11.24 m) the agreement between two models (11) and (14) in general is good for the mean velocity profile as well as for the mean temperature. In a case of a stable stratification (L=112.43 m, L=22.49 m) the profiles computed on two models are agreed only for and then the profiles diverge one from another. These outcomes can be explained with the next simple model. Put the new variable

in the third equation (1.14) normalised as

,

and then it can be rewritten as

.

Using the approximate formula

, where ,

which can be realised at a large distance from the wall (as it has been established in numerical experiments it can be for ), one can transformed this equation to the form

,

where . Put for the mean flow, then integrating the derived equation we have

,

,

where .

Figure 6 The exponent b of the mean velocity profile in the stratified turbulent boundary layer versus the stability parameter 1/L. The symbols were obtained in the numerical experiments; the solid line is calculated on (16)

Using the asymptotic formula

for

and put , the last equation can be written as follows

(15)

But the mean velocity profile is described by logarithmic function for therefore for we have

where .

As it has been reported by Lui & Kotoda [120] the typical value of the exponent in the velocity profile (15) varies for the atmospheric surface layer from for the unstable stratification up to for the stable stratified flows. This is in a good agreement with the numerical data - see Figure 6. The numerical data shown in Figure 6 can be approximated as follows

(16)

In a case of stable stratification , thus the mean velocity profile increases with the distance from the ground surface as a power function with exponent dependent on the stability parameter accordingly to (16). In a case of unstable stratification , hence the mean velocity decreases with the distance as a power function. In a case when the mean velocity profile is resulted to the logarithmic function, that is connected with a neutral stratified flow.

This can be proved also for the mean temperature profile which can be described for by a power function

where , , and .

Turbulent boundary layer in pressure gradients

The boundary layer can be characterized by a velocity distribution on the upper bound of the turbulent flow, which is connected to a pressure gradient by virtue of the Bernoulli theorem:

. (17)

Though on the boundary of a turbulent layer acceleration and pressure gradient are connected, nevertheless inside a layer this connection is upset. In a fluid flow the pressure depends on the force affixed to a chosen surface. Therefore, if pressure is affixed on the bound of a boundary layer, it does not completely mean, as the layer as a whole is under effect of this force. The acceleration is connected with a volumetric force, applied to the unit of volume. Therefore the turbulent boundary layer as a whole can be characterized by the external acceleration, that follows immediately from the principle of relativity. Nevertheless, in the literature [121-124] and other the nomenclature has already been taken in connection with the description of turbulent boundary layers in favourable and adverse pressure gradients. So, we will use this nomenclature in a sense, determined by equation (17).

Let us consider the problem about an accelerated turbulent flow in the boundary layer with a mean acceleration defined as

where is the constant which can be estimated from the experimental data. The equation system for the mean flow in the turbulent boundary layer in pressure gradient can be derived directly from (2.10). Put

in this equation, and then we have

(18)

,

where

is the dimensionless acceleration parameter.

The non-linear subsystem included first and second equation (18) is independent on the streamwise mean acceleration. Thus it can be supplemented by the boundary conditions in the form (2.20) to integrate this subsystem. Substituting the non-linear subsystem solution in the third equation (18) we can write the fist integral of this equation as follows

(19)

Here is the integral computed as in the case of the turbulent flow in a zero pressure gradient:

Here is the solution of the problem (2.19)-(2.20).

Note, that the integral term in the right part of the second equation (19) can be transformed as follows

This integral diverges as the logarithmic function at , since for . It means that the acceleration effects on the turbulent flow in the outer region more intensively then in the inner layer. In the normal turbulent boundary layer (far away from the verge of separation) the acceleration parameter is a small quantity, i.e., . In this case, the inner layer is formed under the same conditions, as in a zero pressure gradient flow.

Using the inner layer variables, the acceleration parameter, and mixed layer parameters, we propose the mean velocity gradient in the turbulent decelerating flows as follows

(20)

As it has been established in numerical experiments, the acceleration parameter with star is given by

,

Hence the acceleration length scale parameter is proportional to the scale of viscous sublayer . Apparently it means that the main part of a power of the supplied pressure gradient is dissipated in the viscous sublayer closed to the wall, and not in the transition layer where .

The mean velocity profile as a solution of the equation (20) depends on the parameter , which can be estimated from the comparison with the experimental data - see Figure 7. The data base by Nagano et al [67-68] has been used to calculate the mean velocity profiles in the turbulent boundary layers in adverse pressure gradients. The estimated profiles are shown in Figure 7. The input data and the numerical parameter for these profiles are listed in the Table 1. The parameters of the mean velocity profile in the turbulent boundary layer in zero pressure gradient are given by

.

The agreement between the predicted mean velocity profile (20) and the experimental data by Nagano et al [67-68] in general is good.

Figure 7 The mean velocity profiles in the turbulent boundary layer in adverse pressure gradients computed on (20) - the solid lines and experimental data from the data base by Nagano et al [68]

The parameter of the mean velocity profile (20) increases with the dimensionless pressure gradient parameter,

,

in a case of adverse pressure gradient - see Figure 8. This function, , can be approximated by a cubic polynomial (the trend line in Figure 8):

.

In a case of strong deceleration the turbulent boundary layer depth grows up to the verge of separation (see chapter 6). Opposite, the accelerated turbulent boundary layer becomes thinner and, eventually, the transition to the laminar flow can happened [124].

Figure 8 The calculated mean velocity profile parameter versus the pressure gradient parameter. The equation of the trend line is shown in the bottom part

Table 1 The input data for the mean velocity profile (20)

Parameter	The cinematic viscosity 105 m2/s	The friction velocity u, m/s	The boundary layer depth H99, mm	The pressure gradient parameter p+	The mean velocity profile parameter
Figure
1.7, b	1.574	0.481	13.3	0	0.79
3.7.1	1.536	0.39	16.2	0.00898	1.0
3.7.2	1.576	0.307	26	0.0181	1.4
3.7.3	1.551	0.251	32	0.023	1.75
3.7.4	1.537	0.197	46.1	0.0251	2.1

Thus, the range of feasibility of the model (20) depends on this two physical phenomena. It is clear, that model (20) cannot be used without the essential modification in the case of the significant acceleration of the turbulent flow.

Nevertheless in the investigated range of parameters model (20) can be agreed well with known experimental data, and some results can be explained with this model. For instance, as it was repeatedly underlined by the authors [67-68], the mean velocity profile in the logarithmic layer in adverse pressure gradients passes lower, than in the turbulent flow in a zero pressure gradient. This outcome immediately follows from the equation (19), which can be presented as

 (21)

Analysing this equation one can conclude that for the turbulent boundary layer in adverse pressure gradient, when , the non-dimensional mean velocity gradient has a less value then in a zero pressure gradient. In a case of favourable pressure gradients, when , the non-dimensional mean velocity gradient has more high value then for a zero pressure gradient flow.

The physical realisation of the accelerated turbulent boundary layers, as it has been explained by Kline et al [70], usually is connected with flows in the convergent channels, around cylinder, in a throat nozzle, etc. In these cases the flow geometry affects on the turbulent boundary layer parameters as well as the flow acceleration. It means that in the second equation (18) can be added some term described the inertial forces acting in the normal to the wall direction, like the buoyancy forces term in the right part of the second equation (11).

The mean velocity profile (20) can be regularised in the mixed layer as follows (see [47]):

(22)

where is given by equation (20). As it has been established the estimated free parameter is independent on this regularisation.

The turbulent boundary layer in adverse pressure gradient develops up to the verge of separation. Close to the verge of separation the skin friction coefficient decreases down to zero. Due to the strong deceleration the back flow is formed in the wall region. The back flow is characterized by the extremely low Reynolds stresses. The logarithmic layer thickness decreases monotonically up to the verge of separation, thus the turbulent separated flow mainly is similar to the mixed layer. The streamwise velocity pulsation profile in the turbulent separated flow has a maximum in the middle part of the mixed layer while in the turbulent flow with the logarithmic layer the maximum of the streamwise turbulent intensity profile situates near the wall (see Figure 2.6). Using the equation system (2.16) the mean velocity profile in the turbulent separated flow can be modelled as well as in the normal turbulent flow.

Planetary boundary layer

One of examples of the theory of turbulence application is the atmospheric boundary layer, which is under effect of a pressure gradient and inertial forces, stipulated by rotation of a planet. It is usually supposed, that in the external flow these forces are counterbalanced, therefore the geostrophic flow is formed, which is described by equation

. (23)

where , is the earth's rotation rate, is the latitude, is the unit vector in OZ direction, is the geostrophic velocity vector.

Hence, in this case it is possible to consider, that the boundary layer is formed under influence of the mean acceleration

.

The non-dimensional acceleration parameters are given by

,

where is the geostrophical wind angle.

Since the atmospheric boundary layer includes the surface layer and the mixed layer, thus the mean velocity profile can be written as a sum of two terms described the turbulent flow in the bottom and top part of the boundary layer. The main model of the rotating flow with arbitrary stratification in the bottom layer is proposed as

(24)

Here is a function of the stability parameter computed on the base of model (11). Note that the model (24) is a generalised form of equation (19). The mean velocity profile as a combination of two terms can be written by analogue to the mean velocity profile in pressure gradient, see (22):

(25)

where

,

is the roughness layer constant. To test this model the velocity profile (25) has been compared with the Leipzig data presented by Detering & Etling [125] and used also by Apsley & Castro [44]. The input data for the wind profile and the constants of model are put as follows:

1) ;

2) .

In the first position the input data, and in the second position the estimated theoretical constants are given. The mean position of the mixed layer is not equal to this value for the turbulent boundary layer on the flat plate. The estimated acceleration parameter, , can be used to calculate the characteristic length scale as

where is the Coriolis parameter (for the Leipzig data). This value can be compared with the maximal mixing length estimated for the Leipzig data by Apsley & Castro [44] as . Hence, if , then .

The transversal component of the flow velocity in the atmospheric boundary layer remains minor up to height about , and then increases. The velocity vector is turned to the right side counting from the local flow direction. To estimate the transversal velocity gradient the experimental Leipzig data shown in Figure 9 can be used. As it follows from this data the turning angle depends on the height approximately as a linear function, . Using the equation connected the turning angle and the wind velocity components, , one can derive the transversal velocity gradient as:

In the logarithmic layer this equation has a simple form:

The normal depth of the atmospheric surface layer is about 1 km, hence the dynamic Reynolds number can be estimated as . Therefore, the transversal velocity gradient has a small value, which can be neglected in the inner layer.

Thus the wind velocity vector is turned primary in the mixed layer under the pressure gradient effect. Due to this the solution for the transversal velocity profile can be written as follows

(26)

,

where . As it has been established the mixed layer parameters in the expression (26) is the same as for the turbulent flow over a flat plate, and all another parameter as in the expression (25), thus

.

The difference in the determination of the mixing layer parameter for the wind velocity components can be explained, first of all, by the fact that the turbulent flow in the atmospheric surface layer varies with time so that the acceleration due to the daily wind velocity variations,, approximately equals to the acceleration due to the planet rotation, . Therefore, the wind velocity profile is the fragment of a solution dependent on time.

The second reason is that the linear form of the equation system (2.12) has been used, to derive the mean velocity component profiles (25)-(26), as follows (see also [47])

(27)

where

is the pressure parameter in the outer region,

, .

The equation system (27) can be integrated in common case as

(28)

where are constants which can be calculated from the inner layer problem. This solution has been used to recover the profiles (25) - (26) in the upper layer. Hence, as follows from (27), the real Coriolis forces has been replaced by the pressure gradient effect.

Nevertheless the model (25)-(26) can be used for the wind profile estimation in the atmospheric boundary layer. The mean velocity profile and the turning of wind with height computed on the model (25)-(26) are shown in Figure 9 - the solid lines, in comparison with the Leipzig data - the black points. The turning angle and the wind velocity are defined as follows , .



Figure 9 Turning of wind with height (left) and the wind velocity profile (right) in the neutral stratified atmospheric boundary layer computed on (25)-(26) - the solid lines, and the Leipzig experimental data [125]

As it has been established, the computed profiles depend on the "infinite" point in which the values are defined. The best correlation with the Leipzig data has been obtained for . This upper bound has also been used for the atmospheric boundary layer parameters estimation by Apsley & Castro [44]. This dependence means that the atmospheric boundary layer can be influenced by the upper layer in a common case including the neutral stratification.

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