# An Analytically Derived Shear Stress and Kinetic Energy Equation for One-Equation Modelling of Complex Turbulent Flows

## Abstract

**:**

## 1. Introduction

_{1}is a constant to be determined and ρ is air density. Since the underlying physics of this equation had been examined by Dryden [3] who found it to be quite valid, the physical basis of Equation (1) could be considered established for the Reynolds number (Re) range of the atmospheric data used to establish Equation (1). The turbulent kinetic energy equation, or k-equation for short, can be derived from the Reynolds stress equations with suitable modelling assumptions invoked for the pressure–velocity correlation and the dissipation rate terms. However, the dissipation rate ε is still not known. Normally, it is determined from the definition of the eddy viscosity which is expressed as a factor multiply by $k/\epsilon $. Thus formulated, the set of equations given by the k-equation, the mean flow equations and Equation (1) would lead to a closed set of equations; therefore, they can be solved using any finite difference scheme. Since dissipation has been assumed isotropic through modelling assumptions, the dissipation rate $\epsilon $ could be expressed in terms of $\left({q}^{2}/2\right)$, the mean flow gradient and an arbitrary constant. Consequently, it gives rise to a one-equation model that could partially account for the history effect of the turbulent kinetic energy because the k-equation solved could account for the convection, diffusion and the production and/or destruction of k [1,3]. This way, the behavior and development of the turbulent shear stress $\tau /\rho $ and $\left({q}^{2}/2\right)$ could be captured more accurately. At this point, it should be pointed out that the analytically derived equations are valid for all Re, while Equation (1) and ${\mathrm{a}}_{1}$ might only be valid for the Re range of the atmospheric data used to deduced Equation (1). Even then, the approach has been adopted by Laster [4], Lee and Harsha [5], Harsha and Lee [6,7,8] in their simulations of free mixing layers and free shear flows. In addition, Harsha and Lee [7,8] determined ${\mathrm{a}}_{1}$ by correlating measurements of $\tau /\rho $ with $({q}^{2}/2)$ for a variety of free turbulent mixing flows. They obtained a value of ${\mathrm{a}}_{1}=0.3$ for Equation (1). These correlations were carried out and used in spite of the fact that Equation (1) was formulated for atmospheric boundary layers where density stratification plays an important role. On the other hand, the free mixing flow data was drawn from simple isothermal flows where external body forces are absent and the Re might be different from the data of atmospheric boundary layer.

#### 1.1. Rationale for Current Study

#### 1.2. Effects of Turbulence Models

#### 1.3. Present Objectives

- (i)
- To show that Equation (1) can be derived from a modelled set of Reynolds stress equations under the assumption of local equilibrium in the near-wall region of a thermal boundary layer. The equation for simple flows with no external body force is derived first; it is then extended to simple and complex flows with different external body forces present.
- (ii)
- To show that ${\mathrm{a}}_{1}$ is a universal constant. It only depends on the set of model constants invoked for the pressure–velocity correlation and the dissipation rate terms. Thus determined, it is also necessary to demonstrate that ${\mathrm{a}}_{1}$ is valid for all turbulent flows with or without external body force effects.
- (iii)
- To show that for flows with external body force effect, the constant ${\mathrm{a}}_{1}$ in Equation (1) would be modified by a function that reflects the importance of the external body force, such as one that depends on the Richardson number of the flow under investigation.
- (iv)
- To show that for complex turbulent flows, the generalized equation has a form similar to Equation (1); however, it could be quite different in details. Thus derived, the resultant one-equation model will also be suitable for simple and complex turbulent flows.

## 2. Basic Equations

## 3. Modelling Assumptions and the Simplified Transport Equations

^{3}axis is assumed to have components (0, 0, $-\mathsf{\Omega}$). If $\left\{{\mathrm{x}}^{\mathrm{i}}\right\}=\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)$ and $\left\{{\mathrm{u}}_{\mathrm{i}}\right\}=\left(\mathrm{u},\mathrm{v},\mathrm{w}\right),$ then it can be shown that

## 4. Case I—Simple Flow

^{2}and $-\overline{\mathrm{uv}}$. The algebra involved is too cumbersome to report here verbatim; however, for the sake of the other cases, the solution procedure could be briefly described below. First, Equations (25)–(27) are solved to give an expression between $-\overline{\mathrm{uv}}$, ${q}^{2}$ and $\partial \mathrm{U}/\partial \mathrm{y}$. The $-\overline{\mathrm{uv}}$ obtained is substituted into Equation (28); and with the help of Equation (26) will yield an expression between ${q}^{2}$ and $\left(\partial \mathrm{U}/\partial \mathrm{y}\right)$, which is Equation (29) given below. Once this is obtained, Equations (26), (28) and (29) can be used to deduce an equation for $-\overline{\mathrm{uv}}$, the final form thus obtained is given in Equation (30).

## 5. Case II—Curved Flow

^{3}or z-axis is assumed to have components (0, 0, −$\mathsf{\Omega}$) for the rotating blade case. In the present case, the y-axis is normal to the x–z plane, while surface curvature defined as K = 1/R also lies on the x–z plane. Therefore, the equations derived for rotating curved flow can be used for the curved flow case by simply setting the rotational speed $\mathsf{\Omega}$ to zero because the blade is stationary. For the present case, i.e., a stationary blade, $\mathsf{\Omega}$ = 0, then the governing equations can be deduced from Equations (18)–(24) by setting $\mathsf{\Omega}$ = 0. In the following equations, the turbulent shear stress $\left(-\overline{\mathrm{uv}}\right)$ is replaced by $\tau /\rho $ and the simplified component equations for the Reynolds stresses, after invoking 2-D boundary layer approximations and assuming production balances dissipation in the constant flux region, can be written as:

## 6. Case III—Rotating Curved Flow

## 7. Case IV—Swirling Flow

_{1}is again given by Equation (32). In addition, ${\mathrm{Ri}}_{\mathrm{S}}$, defined in Equation (61), has the meaning of a gradient Richardson number and is the correct parameter to use to characterize swirling flows. Therefore, for swirling flows, relations similar to Equation (1) are given by Equations (58a) and (59a) for the turbulent shear stresses ${\tau}_{\mathrm{rx}}\mathrm{and}{\tau}_{\mathrm{r}\varphi}$, respectively. In the swirling flow case, the a

_{1}modifying functions given in Equations (58a) and (59a) assume a more complicated form compared to the simple form seen in the curved flow and rotating curved flow cases. This means that the complexity of the modifying function in the equivalent Equation (1) is greatly influenced by the complexity of the body forces, which, in turn, is influenced by the interactions of the 3-D mean field with the 3-D turbulence field. However, for the curved flow and rotating curved flow cases, the mean field is still 2-D in nature; thus, the modifying functions are relatively simple in those cases. The ratio of the shear stresses ${\tau}_{\mathrm{r}\varphi}/{\tau}_{\mathrm{r}\mathrm{x}}$ is then given by the ratio of G/H, i.e., Equation (59b) divided by Equation (58b). The result is

## 8. Case V—Atmospheric Boundary Layer

## 9. Discussion

^{2}/2 equation for the five cases are readily available for examination, and the complex external body force effect on the $\tau /\rho $ and k = q

^{2}/2 equation can be easily identified and studied. These results are:

#### One-Equation Model Based on Equation (87)

^{th}component of the mean velocity, ${x}_{j}$ is the j

^{th}component of the coordinate and ${\sigma}_{k}$ is a constant. The unknown dissipation rate ε is determined by invoking the isotropic dissipation assumption and by making use of the turbulent viscosity ${\nu}_{t}$ definition, namely, ${\nu}_{t}=\tau /\left(\partial {U}_{i}/\partial {x}_{j}\right)$. Thus simplified, the result yields $\epsilon ={C}_{\mu}\left({k}^{2}/\tau \right)\left[\partial {U}_{i}/\partial {x}_{j}\right]$ where ${C}_{\mu}$ is a constant. This way, $\epsilon $ can be determined once the shear stress τ is known. The two-equation turbulence model is then reduced to solving Equation (88) with τ given by Equation (87). Since Equation (87) is derived analytically from the simplified Reynolds equations for a wide variety of flows and ${\mathrm{a}}_{1}$ only depends on the model constants invoked, the equation is valid for all Re. Consequently, the questions posed in Section 1.1 have all been answered and the one-equation model thus formulated is valid for all Re, much like other turbulence models, such as the zero-equation, one-equation, two-equation and Reynolds stress models, discussed in [1]. The k-equation is relatively simple and easy to implement. Further, unlike other one-equation models [1], the k-equation in this one-equation model can also account for the history effect of the turbulent kinetic energy because the modelled k-equation does account for the convection, diffusion and the production and/or destruction of k. Demonstration of the validity and viability of this one-equation approach to model turbulent flows with no body force effects had already been provided by a host of researchers [4,5,6,7,8,9,10,11,12,13,14]; consequently, its extension to complex turbulent flows with body force effects provides an attractive alternative to the conventional more complicated two-equation models.

## 10. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\mathrm{a}}_{1}$ | constant in Equation (1) |

A | constant in Equation (87) |

${\mathrm{g}}_{i}=\left(0,0,-\mathrm{g}\right)$ | gravitational vector |

$\mathrm{g}$ | gravitational constant |

k = ${q}^{2}/2$ | turbulent kinetic energy |

K = 1/R | surface curvature |

p | fluctuating static pressure |

P | mean static pressure |

${\mathrm{Pr}}_{\mathrm{t}}$ | turbulent Prandtl number for stationary plane flow |

${q}^{2}/2=\overline{{\mathrm{u}}^{2}}+\overline{{\mathrm{v}}^{2}}+\overline{{\mathrm{w}}^{2}}$ | kinetic energy of turbulence |

r | radial distance from origin of co-ordinate system |

R | radius of curvature or distance from the axis of rotation |

Re | flow Reynolds number |

${\mathrm{Ri}}_{\mathrm{B}}$ | Richardson number for atmospheric boundary layer |

Ri_{C} | Richardson number for curved flow |

${\mathrm{Ri}}_{\mathrm{F}}$ | Richardson number for complex flow |

${\mathrm{Ri}}_{\mathrm{RC}}$ | Richardson number for rotating curved flow |

${\mathrm{Ri}}_{\mathrm{S}}$ | Richardson number for swirling flow |

T | mean temperature of flow |

${\mathrm{U}}_{\mathrm{j}}$ | mean velocity vector |

U | mean velocity along x-axis |

V | mean velocity along r-axis or y-axis |

W | circumferential mean velocity along the ϕ direction |

$\mathrm{u},\mathrm{v},\mathrm{w}$ | fluctuating velocities along x, y, z or z, r, ϕ directions, respectively |

x, y, z | coordinates along the flow, normal to the flow, and normal to the x–y plane, respectively |

## Greek Symbols

$\alpha $ | the kinematic heat conductivity (or thermal diffusivity) |

$\mathsf{\beta}=72\mathsf{\gamma}/\left(1-6\mathsf{\gamma}\right)$ | function introduced in Equation (38) to abbreviate equation writing |

${\mathsf{\beta}}_{1}$ | constant defined in Equation (77) |

$\mathsf{\gamma}={\ell}_{1}/{\mathsf{\Lambda}}_{1}$ | length scale ratio |

${\mathsf{\gamma}}_{1}={\ell}_{2}/{\mathsf{\Lambda}}_{1}$ | length scale ratio introduced in Equation (76) |

${\mathsf{\gamma}}_{2}={\mathsf{\Lambda}}_{2}/{\mathsf{\Lambda}}_{1}$ | length scale ratio introduced in Equation (76) |

${\sigma}_{\mathrm{T}}$ | molecular thermal diffusivity |

${\u03f5}_{ijk}$ | alternating tensor |

$\epsilon $ | local turbulent dissipation rate |

$\mathsf{\theta}$ | fluctuating temperature |

$\kappa =-\frac{1}{\rho}{\left(\frac{\partial \rho}{\partial T}\right)}_{p}$ | coefficient of thermal expansion |

${\ell}_{1}$ | length scale introduced in Equation (10) |

${\ell}_{2}$ | length scale introduced in Equation (11) |

${\mathsf{\Lambda}}_{1}$ | dissipation length scale introduced in Equation (12) |

${\mathsf{\Lambda}}_{2}$ | dissipation length scale introduced in Equation (73) |

$\rho $ | fluid density |

$\nu $ | fluid kinematic viscosity |

${\nu}_{\mathrm{t}}$ | eddy viscosity or turbulent kinematic viscosity |

$\tau $ | turbulent shear stress |

$\mathsf{\Omega}$ | rotational speed of co-ordinate system about z-axis |

## References

- Speziale, C.G.; So, R.M.C. Turbulence modelling and simulation. In The Handbook of Fluid Dynamics, 1st ed.; Johnson, R.W., Ed.; CRC Press LLC: Washington, DC, USA, 1998; Chapter 14; pp. 4-1–4-111. [Google Scholar]
- Nevzgljadov, V. A Phenomenological Theory of Turbulence. J. Phys.
**1945**, 9, 235–243. [Google Scholar] - Dryden, H.L. Advances in Applied Mechanics; Academic Press: New York, NY, USA, 1948; Chapter 1; pp. 1–40. [Google Scholar]
- Laster, M.L. Inhomogeneous Two-Stream Turbulent Mixing Using the Turbulent kinetic energy equation. In Inhomogeneous Two-Stream Turbulent Mixing Using the Turbulent Kinetic Energy Equation; Defense Technical Information Center: Fort Belvoir, VA, USA, 1970. [Google Scholar]
- Lee, S.C.; Harsha, P.T. Use of turbulent kinetic energy in free mixing studies. AIAA J.
**1970**, 8, 1026–1032. [Google Scholar] [CrossRef] - Harsha, P.T.; Lee, S.C. Analysis of coaxial free mixing using the turbulent kinetic energy method. AIAA J.
**1971**, 9, 2063–2066. [Google Scholar] [CrossRef] - Harsha, P.T. Prediction of free turbulent mixing using a turbulent kinetic energy method. In Free Turbulent Shear Flows, Conference Proceedings, NASA-SP-321; Langley Research Center: Hampton, VA, USA, 1972; Volume 1, pp. 463–521. [Google Scholar]
- Harsha, P.T.; Lee, S.C. Correlation between turbulent shear stress and turbulent kinetic energy. AIAA J.
**1970**, 8, 1508–1510. [Google Scholar] [CrossRef] - Bradshaw, P.; Ferriss, D.H.; Atwell, N.P. Calculation of boundary-layer development using the turbulent energy equation. J. Fluid Mech.
**1967**, 28, 593. [Google Scholar] [CrossRef] - Bradshaw, P.; Ferriss, D.H. Calculation of boundary-layer development using the turbulent energy equation: Compressible flow on adiabatic walls. J. Fluid Mech.
**1971**, 46, 83–110. [Google Scholar] [CrossRef] - Rodi, W.; Mansour, N.N.; Michelassi, V. One-Equation Near-Wall Turbulence Modeling with the Aid of Direct Simulation Data. J. Fluids Eng.
**1993**, 115, 196–205. [Google Scholar] [CrossRef] - Morel, T.; Torda, T.P.; Bradshaw, P. Turbulent kinetic energy equation and free mixing. In Free Turbulent Shear Flows, Conference Proceedings, NASA-SP-321; Langley Research Center: Hampton, VA, USA, 1972; Volume 1, pp. 549–561. [Google Scholar]
- Baldwin, B.; Barth, T. A one-equation turbulence transport model for high Reynolds number wall-bounded flows. In Proceedings of the 29th Aerospace Sciences Meeting, Reno, NV, USA, 7–10 January 1991. [Google Scholar] [CrossRef]
- Spalart, P.; Allmaras, S. A one-equation turbulence model for aerodynamic flows. Rech. Aerosp.
**1994**, 1, 5–21. [Google Scholar] - Oberlack, M.; Wacławczyk, M.; Rosteck, A.; Avsarkisov, V. Symmetries and their importance for statistical turbulence theory. Mech. Eng. Rev.
**2015**, 2, 15–00157. [Google Scholar] [CrossRef] - So, R.M.C. Effects of streamline curvature on the Law of the Wall. In Proceedings of the 12th Annual Meeting of the Society of Engineering Science, New York, NY, USA, 20–22 October 1975; pp. 787–796. [Google Scholar]
- Lai, Y.G.; So, R.M.C. On near-wall turbulent flow modelling. J. Fluid Mech.
**1990**, 221, 641. [Google Scholar] [CrossRef] - Lai, Y.; So, R. Near-wall modeling of turbulent heat fluxes. Int. J. Heat Mass Transf.
**1990**, 33, 1429–1440. [Google Scholar] [CrossRef] - Gerodimos, G.; So, R.M.C. Near-Wall Modeling of Plane Turbulent Wall Jets. J. Fluids Eng.
**1997**, 119, 304–313. [Google Scholar] [CrossRef] - Yang, Z.; Shih, T.H. New time scale-based k-ε model for near-wall turbulence. AIAA J.
**1994**, 31, 1191–1198. [Google Scholar] [CrossRef] [Green Version] - Sarkar, A.; So, R. A critical evaluation of near-wall two-equation models against direct numerical simulation data. Int. J. Heat Fluid Flow
**1997**, 18, 197–208. [Google Scholar] [CrossRef] - Launder, B.; Sharma, B. Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transf.
**1974**, 1, 131–137. [Google Scholar] [CrossRef] - Wilcox, D.C. Turbulence Modeling for CFD, 1st ed.; DCW Industries Inc.: La Cañada, CA, USA, 1993. [Google Scholar]
- So, R.M.C.; Speziale, C.G. A Review of turbulent heat transfer modeling. Annul. Rev. Heat Transf.
**1999**, 10, 177–220. [Google Scholar] [CrossRef] - Mellor, G.L. Analytic Prediction of the Properties of Stratified Planetary Surface Layers. J. Atmos. Sci.
**1973**, 30, 1061–1069. [Google Scholar] [CrossRef] - So, R.M.C. A turbulence velocity scale for curved shear flows. J. Fluid Mech.
**1975**, 70, 37–57. [Google Scholar] [CrossRef] - So, R.M.C. Turbulence velocity scales for swirling flows. In Turbulence in Internal Flows; Murthy, S.N.B., Ed.; Hemisphere Publishing Corp.: London, UK, 1977; pp. 347–369. [Google Scholar]
- Rotta, J. Statistische Theorie nichthomogener Turbulenz. Eur. Phys. J. A
**1951**, 129, 547–572. [Google Scholar] [CrossRef] - Kolmogorov, A.N. The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Cr Acad. Sci. URSS
**1941**, 30, 301–305. [Google Scholar] - Kolmogorov, A.N. The equations of turbulent motion in an incompressible fluid. Izv. Akad. Nauk SSSR
**1942**, 6, 1–2. [Google Scholar] - So, R.M.C. Reynolds analogy and turbulent heat transfer on rotating curved surfaces. Heat Mass Transf.
**2020**, 56, 1–18. [Google Scholar] [CrossRef] - Monin, A.S.; Oboukhov, A.M. Basic turbulent mixing laws in the atmospheric surface layer. Trudy. Geofiz. Inst. An. SSSR
**1954**, 24, 163–187. [Google Scholar] - So, R.M.C.; Mellor, G.L. Turbulent boundary layers with large streamline curvature effects. J. Appl. Math. Phys. ZAMP
**1978**, 29, 54–74. [Google Scholar] [CrossRef] - Hwang, B.C. A Simulation Model of the Planetary Boundary Layer at Kennedy Space Centre; NASA CR-145357; NASA: Washington, DC, USA, 1978. [Google Scholar]

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So, R.M.C.
An Analytically Derived Shear Stress and Kinetic Energy Equation for One-Equation Modelling of Complex Turbulent Flows. *Symmetry* **2021**, *13*, 576.
https://doi.org/10.3390/sym13040576

**AMA Style**

So RMC.
An Analytically Derived Shear Stress and Kinetic Energy Equation for One-Equation Modelling of Complex Turbulent Flows. *Symmetry*. 2021; 13(4):576.
https://doi.org/10.3390/sym13040576

**Chicago/Turabian Style**

So, Ronald M. C.
2021. "An Analytically Derived Shear Stress and Kinetic Energy Equation for One-Equation Modelling of Complex Turbulent Flows" *Symmetry* 13, no. 4: 576.
https://doi.org/10.3390/sym13040576