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This research used the common decomposition of the velocity and pressure in an average part and a fluctuating part, for high Reynolds number, of the Navier-Stokes equation, which leads to the classic problem of turbulent closure. The Prandtl’s mixing length model, based on the Boussinesq hypothesis and traditionally used for free shear flows, was chosen and adapted for internal flows to solve the closure problem. For channel flows, Johann Nikuradse proposed a model for the Prandtl mixing length. In the present paper, which has an academic character, the authors made a return to the model of the mixing length of Prandtl and the model of Nikuradse, to solve turbulent flows inside a plane channel. It was possible to develop an ordinary differential model for the velocity in the direction of the flow whose solution occurs computationally in a simple but extremely accurate way when compared with Direct Numerical Simulation databases. For the viscous stress on the wall , it was possible to determine the exact mathematical solution of the ordinary differential equation. It is a model of great academic value and even to be used as reference for verification of computational codes destined to the solution of complete numerical and computational models.

Turbulent flows are more common in nature and in industrial applications when compared to laminar flows. It is possible to determine exact and numerical solutions for laminar flows. However, for transition to turbulence and for fully turbulent flows, there are no exact solutions, except for some cases of canonical flows. Due to the high complexity and, at the same time, its high practical importance, this subject has been the object of great efforts of the scientific community around the world.

Due to the great academic and industrial interest related to this theme, many concepts and methods were developed to model the average characteristic information of turbulent flows, incorporating the effects of the turbulence itself, Boussinesq (1877) [

Studies involving exact solutions of canonical turbulent flows, such as, free shear flows, can be found in the literature, Schlichting [

Numerous works are found about turbulence closure modeling, such as the Prandtl mixing length model [

It is important to emphasize that for internal flows, in the near regions of walls, the viscous molecular effects grow dramatically as it approaches the walls, which dampens the effects of the turbulence. The model of the mixing length of Prandtl, by itself, does not correctly model this physical characteristic of the internal flows. Nikuradse [

Poiseuille laminar and turbulent flows in flat and circular channels are among the first flows that were solved by the scientific community. Initially this type of flow was solved in laminar regime by Poiseuille (1846) [

Laminar and turbulent, isothermal flows of Newtonian fluids can be modeled by the continuity and Navier-Stokes equations. These equations, for incompressible flows, when submitted to the mean temporal operator, are written below, using indicial notation:

∂ u ¯ i ∂ x i = 0. (1)

∂ u ¯ i ∂ t + ∂ u ¯ i u ¯ j ∂ x j = − 1 ρ ∂ p ¯ ∂ x i + ∂ ∂ x j ( ν ∂ u ¯ i ∂ x j − u ′ i u ′ j ¯ ) . (2)

Equation (1) represents the mean mass balance over a fluid particle, while Equation (2) represents the linear momentum balance over the same fluid

particle. These differential equations can be simplified by the assumption of Poiseuille flows, i.e.: v ¯ = w ¯ = 0 , ∂ p ¯ / ∂ x ≠ 0 and ∂ u ¯ / ∂ x = 0 . Finally, the average flow is considered to be in steady state. Thus, Equation (3) and Equation (4) are obtained:

∂ u ¯ ∂ x = 0. (3)

1 ρ ∂ p ¯ ∂ x = ∂ ∂ y ( ν ∂ u ¯ ∂ y − u ′ v ′ ¯ ) . (4)

It is observed that the Equation (4) presents an additional term u ′ v ′ ¯ that is not known. This term makes the model mathematically open. This fact requires an additional equation for the closure of the model.

The decomposition of velocities and pressure leads to the appearance of the Reynolds stress tensor, which, for the three-dimensional formulation, is given by Equation (5):

τ = [ u ′ u ′ ¯ u ′ v ′ ¯ u ′ w ′ ¯ v ′ u ′ ¯ v ′ v ′ ¯ v ′ w ′ ¯ w ′ u ′ ¯ w ′ v ′ ¯ w ′ w ′ ¯ ] . (5)

Boussinesq proposed to model this tensor using his hypothesis, which consists of making an analogy to the Stokes model for the molecular viscous stress tensor. The model is represented by Equation (6), for one-dimensional flows, as is the case of the flow analyzed in the present work:

− u ′ v ′ ¯ = ν t ( d u ¯ d y + d v ¯ d x ) − 2 3 k δ x y (6)

where k = 1 / 2 ( u ′ u ′ ¯ ) is called turbulent kinetic energy, δ x y is the Cronecker’s operator, and ν t was defined as the kinematic turbulent viscosity. While molecular viscosity is a physical property of fluids, turbulent viscosity is a property of the flow. Substituting this model into Equation (4) and simplifying terms, results in Equation (5):

d d y [ ( ν + ν t ) d u ¯ d y ] = 1 ρ d p ¯ d x . (7)

The turbulent viscosity modeling is presented in the following section.

The turbulent viscosity that appears in Equation (7) remains to be modeled and calculated. One of the first proposals for this was made by Prandtl [

ν t = l m 2 | d u ¯ | | d y | . (8)

Substituting Equation (8) into Equation (7) gives equation (9):

d d y ( ν d u ¯ d y + l m 2 | d u ¯ | | d y | d u ¯ d y ) = 1 ρ d p ¯ d x . (9)

Analyzing

| d u ¯ | | d y | = − d u ¯ d y (10)

Equation (10) is valid for any value of the coordinate 0 ≤ y ≤ R . Substituting Equation (10) into Equation (9) we obtain Equation (11), which is written in a form suitable to be solved.

d d y [ ν d u ¯ d y − l m 2 ( d u ¯ d y ) 2 ] = 1 ρ d p ¯ d x . (11)

where d p ¯ d x is a constant, the term l m is called Prandtl’s mixing length and its modeling varies depending on the flow. This method is used for solving mixing layer flows, and the most basic solution is to consider l m linear. Usually it is assumed as l m = β x , where b is a constant that depends on the flow, where every value is obtained empirically, and shown by Silveira [

Even though linear l m gives a satisfactory result for free-shear flows, like mixing layer, for internal flows, inconsistent results are generated. Since the molecular viscous effects are more relevant near the wall than the turbulent effects, Nikuradse proposed a polynomial empirical equation in [

L ( y ) = l m R = 0.14 − 0.08 ( y R ) 2 − 0.06 ( y R ) 4 . (12)

Now, Equation (11) can be rewritten using Equation (12):

d d y [ ν d u ¯ d y − L 2 ( y ) R 2 ( d u ¯ d y ) 2 ] = 1 ρ d p ¯ d x . (13)

Considering that Equation (13) is a second order differentiation equation, its solution requires two boundary conditions. On the walls the non-slip condition must be applied by Equation (14). In the center of the channel, a second specie boundary condition of Neumann, that is, a condition of symmetry was used, given by Equation (15).

u ¯ ( R ) = 0. (14)

d u ¯ ( 0 ) d y = 0. (15)

Integrating Equation (13) we have:

∫ d [ ν d u ¯ d y − L 2 ( y ) R 2 ( d u ¯ d y ) 2 ] = 1 ρ d p ¯ d x ∫ d y ⇒ ν d u ¯ d y − L 2 ( y ) R 2 ( d u ¯ d y ) 2 = 1 ρ d p ¯ d x y + c . (16)

Applying the boundary condition given by Equation (15) we obtain:

ν ⋅ 0 − L 2 ( y ) R 2 ( 0 ) 2 = 1 ρ d p ¯ d x ⋅ 0 + c ⇒ c = 0. (17)

Now, rearranging Equation (16), we have the following second order polynomial equation in d u ¯ d y :

L 2 ( y ) R 2 ν ( d u ¯ d y ) 2 − d u ¯ d y + 1 ρ ν d p ¯ d x y = 0. (18)

This second-order equation in d u ¯ d y can be solved and two roots can be determined. Only one provides a physically consistent result, which is given by Equation (19):

d u ¯ d y = 1 − 1 − 4 L 2 ( y ) R 2 ν 2 ρ d p ¯ d x y 2 L 2 ( y ) R 2 ν . (19)

In Equation (19), the presence of L ( y ) in the denominator generates indeterminacy, since it can assume null value. To remove this indetermination, the authors proposed the following algebraic manipulation was performed:

d u ¯ d y = ( 1 − 1 − 4 L 2 R 2 ν 2 ρ d p ¯ d x y ) ( 2 L 2 R 2 ν ) ( 1 + 1 − 4 L 2 R 2 ν 2 ρ d p ¯ d x y ) ( 1 + 1 − 4 L 2 R 2 ν 2 ρ d p ¯ d x y ) (20)

d u ¯ d y = 2 y 1 ρ d p ¯ d x ν ( 1 + 1 − 4 L 2 R 2 ν 2 ρ d p ¯ d x y ) .

The Equation (20) is a first order, homogeneous, non-linear ordinary differential equation, with no exact solution. Therefore, a numerical tool is required, and the software MATLAB was used to resolve this equation, using the fourth order Runge-Kutta method already implemented in the software as a function, and well explained in Cheever [

R = 1 m , ρ = 100 kg / m 3 , d p ¯ d x = 100 Pa / m , ν = 10 − 3 m 2 / s .

The velocity distribution, as a function of the y-coordinate, taking into account the conditions given above, is shown in

u ¯ average = 1 R ∫ 0 R u ¯ d y . (21)

Consequently:

R e D = 2 R u ¯ average ν . (22)

Equation (20) was nondiminesionalized, resulting in an equation with a single parameter. This procedure facilitates the comparison of the results obtained in the present study with results of other authors. Let’s introduce the following parameters without dimensions:

y ˜ = y R , u ¯ ˜ = u ¯ u τ , R e τ = u τ R ν . (23)

Two new terms appeared in Equation (23), the friction velocity ( u τ ) and the friction velocity Reynolds number ( R e τ ). These parameters are quite used in the study of turbulent channels and ducts. The friction velocity is defined by Equation (24):

u τ = | τ w | ρ . (24)

where τ w is the shear stress on the wall, which is illustrated in

p 1 2 R = 2 τ w Δ x + p 2 2 R (25)

τ w R = p 1 − p 2 Δ x .

Taken lim δ x → 0 in Equation (25) and using Equation (24), we obtain:

− d p ¯ d x = τ w R = u τ 2 ρ R . (26)

Using these parameters, Equation (20) can be transformed into Equation (27).

d u ¯ d y = − 2 y 1 ρ u τ 2 ρ R ν ( 1 + 1 + 4 L 2 R 2 ν 2 ρ u τ 2 ρ R y ) = − 2 y u τ 2 R ν ( 1 + 1 + 4 L 2 R 2 ν 2 u τ 2 R y ) = − 2 y ˜ R e τ u τ R 1 + 1 + 4 L 2 R e τ 2 y ˜ . (27)

Multiplying both sides of Equation (27) by R / u τ we obtain the final nondimensional form:

d u ¯ ˜ d y ˜ = − 2 y ˜ R e τ 1 + 1 + 4 L 2 R e τ 2 y ˜ . (28)

The new nondimensional boundary conditions are given by Equation (29) and Equation (30). Reminding that equation (30) was already applied.

u ¯ ˜ ( 1 ) = 0 , (29)

d u ¯ ˜ ( 0 ) d y ˜ = 0. (30)

Equation (28) requires only one control parameter, the Reynolds number

based on the friction velocity, u τ , in order to characterize the physical flow to be simulated and compared with other authors. Once we obtain the velocity profile we can obtain the R e D , using Equation (21) and Equation (22), as shown below:

R e D R e τ = 2 R u ¯ average ν 1 R e τ = 2 R u ¯ average ν ν u τ R = 2 R ∫ 0 R u ¯ u τ d y = 2 ∫ 0 1 u ¯ ˜ d y ˜ (31)

R e D = 2 R e τ ∫ 0 1 u ¯ ˜ d y ˜ .

To validate the method presented in the present work, DNS data already existent in the literature, was used. The results to be compared are from DNS simulations of a turbulent channel done by JHTDB [

Beginning with R e τ = 180 and R e τ = 395 , the mean velocity profiles are plotted in

For each case it is possible to calculate R e D using Equation (31), therefore: for R e τ = 180 , R e D = 3490 ; R e τ = 395 , R e D = 9230 ; R e τ = 590 , R e D = 14975 ; and R e τ = 1000 , R e D = 28036 . By

That happens because the method used in the present paper is for completely turbulent flows. It is not valid for laminar and transitional flows. Thus, the lower is the Reynolds values, worst is the precision of the method. However, even with high Reynolds number values the results do not agree with DNS data base, since there is still a big difference between the presented curves and DNS curves.

As previously stated, the Prandtl mixing length should tend to zero near the walls, since the viscous effects are more relevant than the turbulent effects. The mixing length function given by Equation (12), effectively zeroes on the walls, but not as fast as it should. This requires the application of an additional damping function. Van Driest (1956) [

l m = 0.4 y [ 1 − e ( − y R R e τ 26 ) ] . (32)

However, Equation (32) is only valid for y ≤ 40 R / R e τ , thus it is necessary to separate the velocity profile in two parts. Cebeci and Bradshaw (1984) [

L ( y ˜ ) = l m R = ( 0.14 − 0.08 y ˜ 2 − 0.06 y ˜ 4 ) { 1 − e [ ( y ˜ − 1 ) R e τ 26 ] } (33)

In

Applying (32) on ODE (28) and solving it, enables to generate the mean velocity profile for the four different R e τ stated before, now calculated with the damping function. Comparison with the DNS results from JHTDB [

The euclidean norm ( L 2 ) and infinity norm ( L ∞ ) was also calculated for a quantitative comparison between the no damping and damping curve, to the DNS. The results are presented in

With the damping equation the profiles and norm practically coincide with the DNS. As presented, for the mixing length without wall function, the higher the Reynolds value, the higher the precision of the method. In the case R e τ = 180 , the R e D is only 6000, which characterize a low Reynolds number turbulence, being out of limit for the use of this method. With the correction of the damping function, the results fit better with the turbulent flow characteristics.

Combining Equations (6), (8) and (10) it is possible to calculate the Reynolds stress, with Equation (34), for y ≥ 0 .

R e τ = 180 | R e τ = 395 | R e τ = 590 | R e τ = 1000 | |
---|---|---|---|---|

No Damping | 0.0792 | 0.0684 | 0.0637 | 0.0567 |

Damping | 0.0200 | 0.0089 | 0.0063 | 0.0043 |

R e τ = 180 | R e τ = 395 | R e τ = 590 | R e τ = 1000 | |
---|---|---|---|---|

No Damping | 0.1511 | 0.1557 | 0.1566 | 0.1567 |

Damping | 0.0429 | 0.0295 | 0.0257 | 0.0212 |

τ = u ′ v ′ ¯ = l m 2 ( d u ¯ d y ) 2 . (34)

Using the parameters in (23), it is possible to obtain the nondimensional Reynolds stress, using the wall stress τ W , It is given by (35):

τ τ W = L 2 ( d u ¯ ˜ d y ˜ ) 2 . (35)

But, Equation (28) gives us the derivative d u ¯ ˜ / d y ˜ exactly, and we can obtain the exact solution for τ / τ W :

τ τ W = − L 2 ( y ) 2 y ˜ R e τ 1 + 4 L 2 ( y ) R e τ 2 y ˜ . (36)

Therefore, using Equation (36) it is possible to plot the Reynolds stress, the same four different values of Reynolds number. Starting with R e τ = 180 (a) and R e τ = 395 (b), plotted in

In

For these case the euclidean norm is also calculated, as well as the infinity norm (

The same analysis performed for the mean velocity profile can be performed here, since it is possible to see that increasing the Reynolds number, the precision of the method is also increased. The closer to full turbulent flow, the closer is the results of the present work as compared with DNS data.

For every Reynolds number, it was possible to notice that the Reynolds stress had the expected behavior. Increasing the Reynolds, more turbulent it became, and consequently higher turbulent stress are obtained. Therefore, as expected, the higher the Reynolds number, the higher the peak of t. Also, as the flow gets more turbulent, the velocity profile flattens, due to the momentum exchange in the y direction. Thus, the viscous region near the wall decreases, and as shown on the plots, the turbulent Reynolds stress peak, moves closer to the wall, as expected.

0.0146 | 0.0095 | 0.0096 | 0.0075 |

0.0644 | 0.0415 | 0.0380 | 0.0398 |

These results show how this theoretical method is reliable, since with only one simple numeric solution by Runge-Kutta for velocity and an exact solution for the shear stress, values and plots can be generated that compares very well to DNS (a method that can take days to run). But it is important to stress that the presented method is limited to this isolated physical problem.

In the present work, one of the first proposals of turbulence closure modeling was used: the mixing length model of Prandtl to solve turbulent flows in a plane channel. The Nikuradse [

A very simple model, composed of an ordinary differential equation, was proposed for the mean velocity of the flow as well as for the viscous stress on the channel walls. The differential equation for the mean velocity was solved using the Runge-Kutta’s integration method. However, the differential equation for viscous shear stress rate was solved exactly. The results obtained for four values of the Reynolds number were compared with the results of Direct Numerical Simulation, and excellent agreement was obtained.

The proposed and solved flow does not have necessarily practical usage, since channel and duct flows are being studied experimentally for several years, with various well-defined tables and results, even for different roughnesses in the walls. However, the main objective of the current study, is for didactical reasons, to demonstrate a resolution of one of the most used models in teaching turbulence courses. The present paper may help students, interested in this scientifical area, to visualize a complete solution of a turbulent flow, for which the laminar regime solution they know since the first contact with fluid dynamics courses.

Another very important aspect is that the method presented in the current paper may be used for validation of numerical and computational methods. The presented model can be easily used with agility as validation reference. To date, the world scientific community has determined the rate of convergence of numerical methods using synthesized solutions for problems of purely mathematical nature. With the model proposed in the present work, one can determine the convergence rate of numerical and computational methods for real turbulent flows in a flat channel.

Antonialli, L.A. and Silveira-Neto, A. (2018) Theoretical Study of Fully Developed Turbulent Flow in a Channel, Using Prandtl’s Mixing Length Model. Journal of Applied Mathematics and Physics, 6, 677-692. https://doi.org/10.4236/jamp.2018.64061