Kepsilon Model
From ThermalFluidsPedia
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{{Turbulence Category}}  {{Turbulence Category}}  
  In the turbulent models presented in the preceding section, the eddy diffusivity, <math>\varepsilon _{M}</math>, was obtained by algebraic expressions. While they are very easy to apply, their shortcoming is their lack of universal applicability because all of them work well only near the wall. As evidenced by  +  In the turbulent models presented in the preceding section, the eddy diffusivity, <math>\varepsilon _{M}</math>, was obtained by algebraic expressions. While they are very easy to apply, their shortcoming is their lack of universal applicability because all of them work well only near the wall. As evidenced by the figures in [[TwoLayer Model]] and [[Van Driest Model]], the agreement between the wall function and the experimental results is not very good for large ''y''<sup>+</sup>. When the above algebraic models are applied to the internal flow, they will break down near the center of the tube. Thus, it is imperative to develop a turbulent model that is applicable in the region that is sufficiently far away from the wall. An ideal turbulent model should be applicable to any region in the turbulent flow although it may not be able to be represented by algebraic equations. Significant advancement of highspeed computers makes developments and applications of nonalgebraic turbulent models possible. Among different nonalgebraic turbulent models, the Kε model is one of the most widely used models and it will be discussed below. 
  The Kε model is based on the analogy between the motion of the fluid packet in the turbulent flow and the random motion of an ideal gas. The kinetic theory of gases stated that the kinematic viscosity of gas can be obtained by <math>\nu =c\lambda /3</math> where c is the mean speed of the molecules and <math>\lambda </math> is the mean free path. For turbulent flow, the kinetic energy due to the fluid packet random motion is  +  
+  The Kε model is based on the analogy between the motion of the fluid packet in the turbulent flow and the random motion of an ideal gas. The kinetic theory of gases stated that the kinematic viscosity of gas can be obtained by <math>\nu =c\lambda /3</math> where ''c'' is the mean speed of the molecules and <math>\lambda </math> is the mean free path. For turbulent flow, the kinetic energy due to the fluid packet random motion is  
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{{EquationRef(1)}}  {{EquationRef(1)}}  
}  }  
  and the mean speed of the fluid packet is  +  and the mean speed of the fluid packet is ''K''<sup>1/2</sup>. The eddy diffusivity for the momentum can be defined in a similar way to the kinematic viscosity of gas <ref name="K1942">Kolmogorov, A.N., 1942, “Equations of Turbulent Motion of an Incompressible Turbulent Fluid,” Izv. Akad. Nauk SSSR, Ser. Fiz., Vol. VI, pp. 5658.</ref>, i.e., 
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{{EquationRef(2)}}  {{EquationRef(2)}}  
}  }  
  where <math>C_{\mu }</math> is a dimensionless empirical constant and L plays the same role as the mean free path in the kinematic viscosity of the gas. Therefore, both K and L must be determined in order to use eq. (  +  where <math>C_{\mu }</math> is a dimensionless empirical constant and ''L'' plays the same role as the mean free path in the kinematic viscosity of the gas. Therefore, both ''K'' and ''L'' must be determined in order to use eq. (2) to evaluate the eddy diffusivity. 
  +  
  +  
  +  
+  If we consider a control volume in a horizontal slender flow region (turbulent boundary layer), balance of the kinetic energy for velocity fluctuation is  
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{{EquationRef(6)}}  {{EquationRef(6)}}  
}  }  
  where eq. (  +  where eq. (2) was used to evaluate the eddy diffusivity. The rate of K destruction, <math>\varepsilon </math>, can be evaluated by analyzing a fluid packet with diameter L, oscillating in the turbulent flow field. If the oscillating velocity is ''K''<sup>1/2</sup>, the drag force acting on the fluid packet will be <math>C_{D}\rho L^{2}(K^{1/2})^{2}</math>, where ''C<sub>D</sub>'' is the drag coefficient that is approximately equal to 1. The mechanical power dissipated per unit mass is 
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{{EquationRef(7)}}  {{EquationRef(7)}}  
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  The Kequation for a boundary layer type flow can be obtained by substituting eqs. (4  +  The Kequation for a boundary layer type flow can be obtained by substituting eqs. (4) – (7) into eq. (2) 
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{{EquationRef(9)}}  {{EquationRef(9)}}  
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  where <math>\sigma _{\varepsilon },\text{ }C_{1},\text{ and }C_{2}</math>are additional dimensional empirical constants. After K and ε are obtained, the eddy diffusivity can be obtained by eliminating L between eqs. (  +  where <math>\sigma _{\varepsilon },\text{ }C_{1},\text{ and }C_{2}</math>are additional dimensional empirical constants. After K and ε are obtained, the eddy diffusivity can be obtained by eliminating L between eqs. (2) and (7), i.e. 
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{{EquationRef(10)}}  {{EquationRef(10)}}  
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  where CD in eq. (  +  where CD in eq. (7) has been set to 1. Equations (8) – (10) become the equations for the Kε model. Both eqs. (8) and (9) are parabolic because diffusion in the xdirection is neglected. Jones and Launder <ref name="JL1972">Jones, W.P., and Launder, B.E., 1972, “The Prediction of Laminarization with a TwoEquation Model of Turbulence,” Int. J. Heat Mass Transfer, Vol. 15, pp. 301314.</ref> suggested the following values for the Kε model: 
<math>C_{\mu }=0.09,\text{ }C_{1}=1.44,\text{ }C_{2}=1.92,\text{ }\sigma _{K}=1,\text{ and }\sigma _{\varepsilon }=1.3</math>  <math>C_{\mu }=0.09,\text{ }C_{1}=1.44,\text{ }C_{2}=1.92,\text{ }\sigma _{K}=1,\text{ and }\sigma _{\varepsilon }=1.3</math>  
  Since the effect of molecular viscosity, ν, was neglected in eqs. (  +  Since the effect of molecular viscosity, ν, was neglected in eqs. (8) and (9), the Kε model cannot be applied in the viscous sublayer where molecular viscosity dominates the velocity profiles. In the sublayer region, Prandtl’s mixing length theory is valid, thus 
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  The generation of K represented by the second term on the righthand side of eq. (  +  The generation of K represented by the second term on the righthand side of eq. (8) becomes 
<math>\varepsilon _{M}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}=\frac{\varepsilon _{M}^{3}}{l^{4}}</math>  <math>\varepsilon _{M}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}=\frac{\varepsilon _{M}^{3}}{l^{4}}</math>  
  Substituting eq. (  +  Substituting eq. (2) into the righthand side of the above equation, one obtains: 
<math>\varepsilon _{M}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}=\frac{C_{\mu }^{3}K^{3/2}L^{3}}{l^{4}}</math>  <math>\varepsilon _{M}\left( \frac{\partial \bar{u}}{\partial y} \right)^{2}=\frac{C_{\mu }^{3}K^{3/2}L^{3}}{l^{4}}</math>  
  Considering eq. (  +  Considering eq. (7), the above equation can be rewritten as 
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<math>L=l\left( \frac{C_{D}}{C_{\mu }^{3}} \right)^{1/4}</math>  <math>L=l\left( \frac{C_{D}}{C_{\mu }^{3}} \right)^{1/4}</math>  
  eq. (  +  eq. (12) becomes 
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Revision as of 02:57, 21 July 2010
External Turbulent Flow/Heat Transfer 
In the turbulent models presented in the preceding section, the eddy diffusivity, , was obtained by algebraic expressions. While they are very easy to apply, their shortcoming is their lack of universal applicability because all of them work well only near the wall. As evidenced by the figures in TwoLayer Model and Van Driest Model, the agreement between the wall function and the experimental results is not very good for large y^{+}. When the above algebraic models are applied to the internal flow, they will break down near the center of the tube. Thus, it is imperative to develop a turbulent model that is applicable in the region that is sufficiently far away from the wall. An ideal turbulent model should be applicable to any region in the turbulent flow although it may not be able to be represented by algebraic equations. Significant advancement of highspeed computers makes developments and applications of nonalgebraic turbulent models possible. Among different nonalgebraic turbulent models, the Kε model is one of the most widely used models and it will be discussed below.
The Kε model is based on the analogy between the motion of the fluid packet in the turbulent flow and the random motion of an ideal gas. The kinetic theory of gases stated that the kinematic viscosity of gas can be obtained by ν = cλ / 3 where c is the mean speed of the molecules and λ is the mean free path. For turbulent flow, the kinetic energy due to the fluid packet random motion is

and the mean speed of the fluid packet is K^{1/2}. The eddy diffusivity for the momentum can be defined in a similar way to the kinematic viscosity of gas ^{[1]}, i.e.,

where C_{μ} is a dimensionless empirical constant and L plays the same role as the mean free path in the kinematic viscosity of the gas. Therefore, both K and L must be determined in order to use eq. (2) to evaluate the eddy diffusivity.
If we consider a control volume in a horizontal slender flow region (turbulent boundary layer), balance of the kinetic energy for velocity fluctuation is

For a twodimensional steady state flow, the convection of K is

Since eddy diffusion in the xdirection is negligible for a boundary layer type flow, only diffusion in the ydirection needs to be considered:

where σ_{K} is a dimensionless empirical constant. The rate of K production can be obtained by the eddy shear stress and the time averaged velocity gradient

where eq. (2) was used to evaluate the eddy diffusivity. The rate of K destruction, , can be evaluated by analyzing a fluid packet with diameter L, oscillating in the turbulent flow field. If the oscillating velocity is K^{1/2}, the drag force acting on the fluid packet will be C_{D}ρL^{2}(K^{1 / 2})^{2}, where C_{D} is the drag coefficient that is approximately equal to 1. The mechanical power dissipated per unit mass is

The Kequation for a boundary layer type flow can be obtained by substituting eqs. (4) – (7) into eq. (2)

By following a similar procedure, the dissipation equation can be obtained as the following:

where are additional dimensional empirical constants. After K and ε are obtained, the eddy diffusivity can be obtained by eliminating L between eqs. (2) and (7), i.e.

where CD in eq. (7) has been set to 1. Equations (8) – (10) become the equations for the Kε model. Both eqs. (8) and (9) are parabolic because diffusion in the xdirection is neglected. Jones and Launder ^{[2]} suggested the following values for the Kε model:
Since the effect of molecular viscosity, ν, was neglected in eqs. (8) and (9), the Kε model cannot be applied in the viscous sublayer where molecular viscosity dominates the velocity profiles. In the sublayer region, Prandtl’s mixing length theory is valid, thus

The generation of K represented by the second term on the righthand side of eq. (8) becomes
Substituting eq. (2) into the righthand side of the above equation, one obtains:
Considering eq. (7), the above equation can be rewritten as

If the length scale is taken as
eq. (12) becomes

where the left and right hand sides, respectively, represent the rates of production and dissipation of K, i.e. generation of K is balanced by dissipation of K in the viscous sublayer. Therefore, the mixing length theory can be used in the sublayer region and the fully turbulent region can be described using the Kε model.
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