CFD of Fully Developed Channel Flow
Jasim M. Jasim1*, Younis M. Najim1, Abdulrahman H. Mohammed 1
1 Department of Mechanical Engineering, Faculty of Engineering, University of Mosul, Mosul, Iraq
*Corresponding Author: [email protected] Accepted: 15 April 2023 | Published: 30 April 2023
DOI:https://doi.org/10.55057/ijarei.2023.5.1.5
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Abstract: Turbulent flow, characterized by random and chaotic fluid motion, is a common occurrence in many engineering applications, such as aerospace, wind turbines, hydroelectric power plants, and chemical mixing processes. In these applications and others, the turbulent flows often involve complex geometries, multiple scales of motion, and nonlinear interactions, which make it challenging to simulate and predict turbulent flow behaviour accurately. As a result, turbulence modelling plays a crucial role in these applications by providing an efficient and cost-effective way to model the complex physics of turbulent flows. Several turbulence models have been developed and are commonly used in computational fluid dynamics (CFD) simulations to predict the behaviour of turbulent flows. In this work, the Navier-Stokes equations and transport equations of turbulent kinetic energy and turbulent dissipation are solved using finite difference approximation. In-house MATLAB code is developed in this work to solve the two-dimensional, incompressible, fully developed turbulent channel flow for Reynolds number ranged from 6000 to 12000. The code has accurately predicted the flow structure in the fully developed turbulent channel flow and for all Reynolds numbers. The velocity, pressure, turbulent kinetic energy, and turbulent dissipation are all aligned with physics and supported by literature data. The turbulent averaged. The averaged turbulent u- velocity profile flattens at the channel core while dramatically changes near wall with power law. The pressure distribution.
Keywords: Turbulent Channel Flow, MATLAB Code, Navier-Stokes Equation, k-ε Turbulent Model
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1. Introduction
Turbulent channel flow refers to the chaotic flow of fluid through a channel, such as a pipe or a duct, where the fluid moves in a disordered, random fashion. This type of flow is characterized by fluctuations in velocity, pressure, and other fluid properties, and is an important area of study in fluid dynamics [1]. Turbulent channel flow is encountered in a wide range of practical engineering applications, including the transportation of fluids in pipelines, the design of heat exchangers, the cooling of electronic equipment, and the production of polymer films. It is important to understand and model turbulent channel flow in order to optimize these processes and prevent issues such as energy loss, clogging, and damage to equipment.
Numerical simulations and computational fluid dynamics (CFD) are used to study turbulent channel flow and to develop models for predicting its behavior. One approach is to use the Reynolds-averaged Navier-Stokes (RANS) equations. Another approach is to use large eddy simulation (LES), which involves filtering the flow field into large and small scales and
simulating the large-scale turbulence while modeling the small-scale turbulence. Direct numerical simulation (DNS) is a third approach, which involves solving the full Navier-Stokes equations without any turbulence models.
Turbulence is unable to support itself and must draw energy from its surroundings. This energy is usually known as turbulent kinetic energy. Shear in the mean flow is a typical source of energy for unpredictable velocity fluctuations. Other sources, such as buoyancy, also can be occurred in the flow. If there is no shear or other preservation mechanism present in the flow, turbulence will decay. As a result, the Reynolds number will drop and the flow will most likely become laminar. Viscous shear stresses cause a distortion that increases the fluid's internal energy at the expense of the kinetic energy of the turbulence. Turbulence needs a continual supply of power to compensate for these viscous losses [2].
One of the earliest studies on turbulent channel flow was performed by Townsend (1951) [3]
who observed that the mean velocity profile in a turbulent channel was logarithmic in nature.
Later, other researchers, such as Perry and Chong (1982) [4] and Kim et al. (1987) [5]
confirmed this logarithmic behavior and provided more detailed insights into the structure of the turbulent flow. More recent studies have focused on the dynamics of the large-scale structures in turbulent channel flow. For example, Smits and Marusic (2013) [6] used a combination of experimental and numerical techniques to investigate the energy budget of the turbulent flow and the role of the large-scale structures in this budget. They found that the large-scale structures were responsible for the majority of the energy transfer between the mean flow and the turbulent fluctuations. Other researchers have focused on the effects of different boundary conditions on turbulent channel flow. For example, Lee and Moser (2015) [7] studied the impact of a spanwise rotation on the flow and found that it had a significant effect on the large-scale structures, which became more organized and coherent in the presence of the rotation.
2. Modeling and Numeric
2.1. Computation domain
The computational domain of this work consists of a two-dimensional area comprising of a channel that has a height of 𝐻 and length of 𝐿. This channel has an inlet on its left and an outlet on its right, as shown in Figure 1. A fluid enters the channel through the inlet from its left direction
Figure 1: Channel flow
2.2 Governing equations (Navier-Stokes Equations)
Navier-Stokes equations describe mathematically the physical relation between the flow variable i.e., velocity, and pressure for incompressible flow, in addition to density for compressible flow. The relation between flow variables is very complicated due to the nonlinearity built in the nature of the Navier Stokes equations, especially the convective term.
The flow velocity is a vector of the three components while the pressure has no directional meaning. For affordable computational cell size, the Navier Stokes solver never converges and sometimes crashes for high Reynolds numbers due to instabilities build in the mathematical and physical nature of the Navier Stokes equations [8]. To eliminate the instabilities arising from the instantaneous fluctuation in flow velocity and pressure, time averaging is introduced such that the instantaneous flow variable is split into the average and fluctuation part (u = u̅ + u′, 𝑝 = 𝑝̅ + 𝑝′). Substitute into the Navier Stokes equations results in what is commonly known as the Reynolds Average Navier Stokes equations (RANS). The RANS has a closure problem that needs to be addressed using turbulence models as will be discussed later. The flow conditions in channel flow are assumed as two-dimensional, steady, incompressible, and constant density. The Navier Stokes equation can be expressed as:
Continuity:
𝜕𝑢̅𝑖
𝜕𝑥𝑖 = 0 (1) Momentum:
𝜕𝑢̅𝑖
𝜕𝑡 + 𝑢̅𝑗𝜕𝑢̅𝑖
𝜕𝑥𝑗= −1
𝜌
𝜕𝑝
𝜕𝑥𝑖+ 𝜕
𝜕𝑥𝑗 (𝜈𝜕𝑢̅𝑖
𝜕𝑥𝑗− 𝑢𝑖′𝑢𝑗′) (2) The Reynolds average of the instantaneous momentum equation introduces The Reynolds stress tensor part (u = u̅ + u′, 𝑝 = 𝑝̅ + 𝑝′) which is the closure problem of momentum’s equation and it’s modeled based on Boussinesq eddy-viscosity approximation such as:
𝑅𝑖𝑗 = 𝜌𝑢𝑖′𝑢𝑗′= 𝜇𝑡(𝜕𝑢̅𝑖
𝜕𝑥𝑗+𝜕𝑢̅𝑗
𝜕𝑥𝑖) −2
3𝜌𝑘𝛿𝑖𝑗 (3)
which can be written in shorthand as 𝑢𝑖′𝑢𝑗′= 𝜈𝑡𝑆𝑖𝑗 −2
3𝑘𝛿𝑖𝑗 (4)
Where
• 𝑆𝑖𝑗 = is the mean rate of strain tensor
• 𝜈𝑡 = nis the turbulent eddy viscosity
• 𝑘 =1
2𝑢𝑖′𝑢𝑗′ turbulent kinetic energy
• 𝛿𝑖𝑗 =is the Kronecker delta function The 2
3𝜌𝑘𝛿𝑖𝑗 term is a normal stress and is analogous to the pressure term that arises in the viscous stress tensor and is typically treated together with the pressure term, by combined it with it. where the used turbulence model in this work is the k-ε model.
2.3 k-ε Model
The k-ε model is the most commonly used two-equation model. The majority of these models resolve a transport equation for turbulent kinetic energy k as well as a second transport equation that enables the definition of a turbulent length scale [8]. Similar transport equations may be created for any other turbulence quantities, such as the rate of viscous dissipation (see Bradshaw et al., 1981) [9]. However, there are several unknowable and unmeasurable terms in the precise ε-equation. The typical k-ε model (Launder and Spalding, 1974) [10]. comprises two model equations, one for k and one for ε. The equations for turbulent kinetic energy and turbulent dissipation are then cast into turbulent eddy viscosity such that:
𝑅𝑖𝑗 = 𝑢̅̅̅̅̅̅ =𝑖′𝑢𝑗′ 1
𝜌 𝜈𝑡 [𝜕 𝑢̅𝑖
𝜕𝑥𝑗 +𝜕 𝑢̅𝑗
𝜕𝑥𝑖] (5) It is worth noted that the eddy viscosity is a kinematic property and has nothing to do with molecular of the medium. The eddy viscosity is then predicted using k-ε model such as:
𝜈𝑡 = 𝐶𝜇𝑘2
𝜀 (6) To create the k-ε model, the initial step is to identify the precise equation for k, The turbulent kinetic energy equation (TKE) is derivate from the Reynolds-averaged Navier-Stokes equations (RANS) by decomposing the velocity field into a mean component and a fluctuating component. (As showing in appendix A) The resulting equation for the evolution of TKE is
𝜕𝑘
𝜕𝑡+ 𝑢̅𝑖 𝜕𝑘
𝜕𝑥𝑗= 𝜈𝑡( 𝜕𝑢̅𝑖
𝜕𝑥𝑗
𝜕𝑢̅𝑗
𝜕𝑥𝑖)𝜕𝑢̅𝑖
𝜕𝑥𝑗− 𝜀 + 𝜕
𝜕𝑥𝑗[(𝜈 −𝜈𝑡
𝜎𝑘) 𝜕𝑘
𝜕𝑥𝑗] (7) The next step involves determining the accurate equation for ε. It is one of the fundamental equations in the k-ε model, The ε-equation is derived by taking the time derivative of the equation for turbulent kinetic energy ε and substituting it in the Reynolds-averaged Navier- Stokes equations. The resulting equation relates the time rate of change of ε to its production, dissipation, and diffusion terms. Once this is completed, the eddy viscosity can be calculated from the turbulent kinetic energy and dissipation, as presented earlier.
𝜕𝜀
𝜕𝑡+ 𝑢̅𝑖 𝜕𝜀
𝜕𝑥𝑗 = 𝜕
𝜕𝑥𝑗[(𝜈 −𝜈𝑡 𝜎𝜀) 𝜕𝜀
𝜕𝑥𝑗] + 𝑐1𝜀2𝜈𝑡𝑆̅𝑖𝑗𝜕𝑢̅𝑖
𝜕𝑥𝑗 𝜀
𝑘− 𝑐2𝜀𝜀2
𝑘 (8) Additionally, the equations include five variable constants are: c_μ, σ_k, σ_ε, c_1ε and c_1ε.
The standard k-ε model uses constant values that were determined by thorough data fitting for a variety of turbulent flows:
𝑐1𝜀=1.44, 𝑐2𝜀=1.92, 𝑐𝜇=0.09, 𝜎𝑘 =1.0, 𝜎𝜀=1.3 The semiempirical method and optimization are used to systematically calculate these coefficients (see, e.g., Rodi, 1980) [11]. The turbulence model that is being used involves solving mean flow conservation equations simultaneously with transport equations in order to determine the turbulence energy and its rate of dissipation. by using a two-dimensional fluid- flow MATLAB code through computational methods.
2.4 Grid generation
The crucial initial stage in computational fluid dynamics involves dividing the domain into small computational cells, which enables the application of mechanical equilibrium conditions to simplify the implementation of fundamental physical principles within the computational domain. In order to accurately capture flow characteristics, the mesh must be fine enough. A collection of cell counters called a i-j set is used in this type of programming, where i stands for the horizontal cell number and j for the vertical cell number. The resulting two-dimensional matrix of zones, sometimes referred to as a MESH, is composed of i x j unique rectangles with dimensions of dx and dy. Figure 2. depicts this mesh
Figure 2: Mesh for channel flow
2.5 Boundary conditions
The boundary conditions of the fluid domain are used to simulate the computational domain are:
• Velocity: The boundary conditions of the fluid domain at the inlet are the u-velocity in (m/s) which is a function of Re as it changes from 6000 to 12000 (𝑢𝑖𝑛=𝑓(Re)), outlet its zero- gradient and the bottom and top its zero due to non-slip condition, also for the v-velocity in (m/s) at inlet, bottom and top which is a equal to zero, due to non-slip condition, in outlet its zero-gradient.
• Pressure: Where the inlet, outlet, bottom and top boundary condition of pressure is assumed that zero-gradient.
• Turbulent kinetic energy and turbulent dissipation rate: The boundary conditions of the fluid domain at the inlet for the TKE and TDR which are assumed that 𝑘 = 0.001𝑚2/𝑠2, 𝜀 = 0.001𝑚2/𝑠3 and in outlet assumed zero-gadient but all the external walls of the solid domains are assumed zero due to non-slip condition near the walls.
3. Results and Discussions
3.1 Two-dimensional velocity field
The two-dimensional axial average velocity in turbulent channel flow refers to the mean velocity profile across a channel. The velocity profile is characterized by a logarithmic region near the wall and a linear region in the core of the channel, as shown in Figure 3. for Reynolds numbers of 4000 to 12000. The logarithmic region is a result of viscous forces at the wall, while the linear region is a result of turbulent mixing in the core of the channel. The thickness of the logarithmic region is proportional to the viscosity of the fluid, while the thickness of the
linear region is proportional to the channel height. In turbulent channel flow. As the Reynolds number increases, the thickness of the logarithmic region decreases, and the slope of the linear region becomes steeper. At low Reynolds numbers, the flow is laminar, and the profile has a parabolic shape. As the Reynolds number increases, the flow becomes more turbulent, and the profile transitions to a logarithmic region near the wall and a linear region in the core of the channel.
Figure 3: Two-dimensional u-velocity for Reynolds numbers of (a) Re=6000, (b) Re=8000, (c) Re=10000, (d) Re=12000.
3.2 Fully developed velocity profile
In fully developed turbulent channel flow, the u-velocity profile shown in Figure 4. is characterized by a flattened parabolic shape with a more uniform velocity distribution across the channel compared to laminar flow. The velocity near the wall is slower than in the center, but the velocity gradient is much steeper than in laminar flow. The velocity profile is influenced by the balance between viscous forces and turbulent fluctuations. The law of the wall describes the velocity profile near the wall as logarithmic, and the thickness of the viscous sublayer is determined by the viscosity of the fluid and the shear stress at the wall. As the Reynolds number increases, the flow becomes more chaotic, and the velocity profile becomes flatter with a more uniform velocity distribution across the channel. The flow experiences frictional forces at the walls of the channel, known as wall shear stress, which cause the fluid near the walls to slow down, resulting in a lower velocity at the wall than in the center of the channel. The velocity in the core region increases to compensate for this reduction and conserve the mass flow rate.
Figure 4: turbulent u-velocity profile for different Reynolds numbers.
The fully developed turbulent channel flow exhibits an interesting profile in the averaged v- velocities, unlike laminar flow see Figure 5. The frictional forces at the channel walls cause the v-velocity to increase as it moves away from the walls, and then it decreases to approach zero at the center line of the channel. After passing the center line, the v-velocity changes direction and produces a reversed symmetrical profile, which is important for turbulent kinetic energy production and dissipation as well as satisfying the continuity equation. Therefore, unlike laminar flow, v-velocity is not zero everywhere in the channel, and it produces two reversed symmetrical profiles about the channel centerline.
Figure 5: Turbulent v-velocity for different Reynolds number
3.3 Pressure field
The two-dimensional pressure field in turbulent channel flow is shown in Figure 6. which refers to the distribution of pressures across the channel's cross-section. Turbulent flow results in pressure fluctuations and vortices within the fluid, generating a complex pressure gradient that is essential to estimating the pumping power required to get the fluid flowing. At moderate Reynolds numbers, the pressure distribution exhibits a characteristic pattern with low pressure regions near the channel walls and high-pressure regions in the center of the channel. As the Reynolds number increases, the pressure distribution becomes more complex, with additional low-pressure regions and high-pressure regions developing in the fluid. Conversely, in laminar flow, the pressure distribution is relatively smooth and uniform across the channel's cross- section. This is clearly shown in Figure 7. where it shows a clear agreement with what we mentioned above [12].
Figure 6: Two-dimensional pressure field for Reynolds numbers of (a) Re=6000, (b) Re=8000, (c) Re=10000, (d) Re=12000.
Figure 7: Distribution of pressure in laminar channel flow for Reynold number of Re=100.
3.4 Turbulent kinetic energy profile
The Figure 8. show the distribution of turbulent kinetic energy in the velocity field within a channel at 2/3 of the channel length for different Reynolds numbers. Turbulent kinetic energy is the energy associated with turbulent fluctuations. At the walls of the channel, the turbulent kinetic energy is zero due to the no-slip condition. However, there is a jump in turbulent kinetic energy very close to the wall, which follows a logarithmic law of the wall. This law states that the turbulent kinetic energy is proportional to the logarithm of the distance from the wall, and
Re=100
this relationship holds up to a certain distance from the wall. The profile of turbulent kinetic energy is nearly universal and independent of flow details. As the Reynolds number increases, the maximum value of turbulent kinetic energy near the wall increases, but the structure remains similar as it moves towards the centerline of the channel with symmetrical behavior about the centerline. This behavior was also observed in previous studies (reference [13]).
Figure 8: Turbulent kinetic energy at 2/3 of the channel length for different Reynolds number.
3.5 Turbulent dissipation
Turbulent dissipation refers to the conversion of turbulent kinetic energy into heat by turbulent eddies. In fully developed turbulent channel flow, turbulent dissipation Figure 9. is highest near the channel walls where the shear is highest, the eddies are strongest, and the velocity gradients are largest. Turbulent dissipation exhibits a symmetrical structure about the channel centerline, similar to TKE. Reynolds number has a significant effect on turbulent dissipation near the wall in a turbulent channel flow. As Reynolds number increases, velocity gradients near the wall increase, resulting in higher turbulence intensity and larger eddies. This leads to an increase in the rate of turbulent dissipation near the wall.
Figure 9: Turbulent dissipation profile at 2/3 of the channel length for different Reynolds numbers.
4.Conclusion
In this study, an in-house MATLAB code was built to solve the two-dimensional, incompressible, fully developed turbulent channel flow. The two-equation k-ε model is used for validation purpose. The inlet velocity is imposed and varied so that Reynolds number varies from 6000 to 12000. The no-slip condition is incorporated in the channel walls and pressure at the outlet has zero-gradient. Several conclusions can be drawn from this research:
The results show that the CFD solver accurately predicts the flow structure and is aligned with physics and literature data.
As the Reynolds number increases in turbulent channel flow the flow becomes more chaotic and the velocity profile becomes flatter, with a more uniform velocity distribution across the channel. This is because the turbulence increases the momentum transfer between the center of the channel and the walls, reducing the velocity gradient near the wall and increasing it in the center of the channel.
The relationship between the Reynolds number and the velocity profile in turbulent channel flow is typically described using the concept of the "inner" and "outer" layers. In the inner layer, which is close to the wall, the flow is strongly influenced by the viscous forces, and the velocity profile is logarithmic, as described by the law of the wall. In the outer layer, which is farther from the wall, the flow is more influenced by the turbulent fluctuations, and the velocity profile becomes more uniform.
The turbulent kinetic energy can be viewed as the distribution of the energy associated with the turbulent fluctuations in the velocity field within the channel. Near channel walls, the turbulent kinetic energy exhibits a dramatic change. On the wall, the turbulent kinetic energy is zero due to no slip condition where both u and v velocities and their fluctuation velocities are zero.
The effect of Reynolds number on the turbulent dissipation near the wall in a turbulent channel flow is significant, as higher Reynolds numbers lead to higher turbulence intensity, larger eddies, and an increase in the rate of turbulent dissipation near the wall.
As the Reynolds number increases further, the pressure distribution becomes more complex, with additional low-pressure regions and high-pressure regions developing in the fluid.
References
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Chalmers University of Technology, 2011.
H. Tennekes, A First Course in Turbulence, Cambridge, Mass: MIT Press,, 1972.
A. A. Townsend, “The structure of the turbulent boundary layer,” Cambridge University Press, vol. 47, no. 2, pp. 375-395, 1951.
A. Perry and M. Chong, “On the mechanism of wall turbulence,” Journal of Fluid Mechanics, vol. 119, pp. 173-217, 1982.
