Turbulent kinetic energy estimate in the near wall region of smooth turbulent channel flows

Abstract

Turbulent kinetic energy (TKE) equation based on the Reynolds averaging hypothesis was approximated in the near wall region of turbulent channel flows and an analytical solution for TKE was proposed by Absi for $y^{ + } < 20$ (R. Absi, Analytical solutions for the modeled k equation, Trans. ASME: J. Appl. Mech. 75 (2008) 044,501). While, comparisons with DNS data show very good agreement, some of the model approximations employed previously require revision which is the main motivation of this work. We here present an alternate formulation for the smooth near wall TKE budget and propose an improved method of solution. Results are compared against wide range of friction Reynolds numbers, Re_τ DNS datasets and found to be in good agreement. Interestingly the proposed improved method reveals that beyond Re_τ, = 5200 TKE profiles collapse meaning that the TKE remains independent of Re_τ that is consistent with high Reynolds number turbulent characteristics reported in the DNS (Hultmark et. al., Turbulent pipe flow at extreme Reynolds numbers, Phys. Rev. Lett., 108 (2012) 094,501). Although there exist two parameters in the paresent work to be determined empirically from DNS data it is reported that these parameters become empirically free at high turbulent Reynolds numbers, Re_τ → ∞.The other interesting aspect of this work is that one of these parameters may be sensitized to other additional effects viz., wall roughness, pressure gradient, magnetic field, etc. One of the potential advantages of these near wall smooth turbulent channel flow analytical solutions is to obviate the need for solving the TKE equation near the smooth wall in the RANS, hybrid RANS/LES closures thus resulting in quicker and accurate engineering computations.

Appendices

Appendix 1

Equation (13) can be rewritten as

$$\frac{\partial }{{\partial y^{ + } }}\left( {C_{{\nu t}} y^{ + } \left( {1 - e^{{ - \frac{{y^{ + } }}{{A_{3}^{ + } }}^{ + } }} } \right)^{2} \frac{{\partial k^{ + } }}{{\partial y^{ + } }}} \right) = \lambda C_{{\nu t}} y^{ + } \left( {1 - e^{{ - \frac{{y^{ + } }}{{A_{3}^{ + } }}^{ + } }} } \right)^{2} \left( {\frac{{\partial U^{ + } }}{{\partial y^{ + } }}} \right)^{2} e^{{\left( { - \frac{{y^{ + } }}{{2\text{Re} _{\tau }^{\alpha } }}} \right)}}$$

(A1)

Retaining only the leading order terms in Eq. (A1), we have

$$\frac{\partial }{{\partial y^{ + } }}\left( {y^{{ + 3}} \frac{{\partial k^{ + } }}{{\partial y^{ + } }}} \right) = \lambda y^{{ + 3}} \left( {\frac{{\partial U^{ + } }}{{\partial y^{ + } }}} \right)^{2} e^{{\left( { - \frac{{y^{ + } }}{{2\text{Re} _{\tau }^{\alpha } }}} \right)}}$$

(A2)

With $U^{ + } = y^{ + }$ and $A_{3}^{ + } = 2\text{Re} _{\tau }^{\alpha }$, Eq. (A2) is written as

$$\frac{\partial }{{\partial y^{ + } }}\left( {y^{{ + 3}} \frac{{\partial k^{ + } }}{{\partial y^{ + } }}} \right) = \lambda \sum\limits_{0}^{\infty } {\frac{{\left( { - 1} \right)^{n} \left( {A_{3} ^{{ + - 1}} } \right)^{3} }}{{n!}}} y^{{ + (n + 3)}}$$

(A3)

Integrating (A3) we get

$$y^{{ + 3}} \frac{{\partial k^{ + } }}{{\partial y^{ + } }} = \lambda \sum\limits_{0}^{\infty } {\frac{{\left( { - 1} \right)^{n} \left( {A_{3} ^{{ + - 1}} } \right)^{3} }}{{n!(n + 4)}}} y^{{ + (n + 4)}} + C1$$

(A4)

where the integration constant C1 = 0 because $\frac{{dk^{ + } }}{{dy^{ + } }} = 0$ at y⁺ = 0. Integrating (A4) we get

$$k^{ + } = y^{{ + 2}} \sum\limits_{0}^{\infty } {\frac{\lambda }{{(n + 2)(n + 4)}}\frac{{\left( { - 1} \right)^{n} \left( {A_{3} ^{{ + - 1}} } \right)^{n} }}{{n!}}} y^{{ + n}} + C2$$

(A5)

where the integration constant C2 = 0 because $k^{ + } = 0$ at y⁺ = 0. Since α appearing in $A_{3}^{ + } = 2\text{Re} _{\tau }^{\alpha }$ is to be determined empirically from DNS, we may safely assume $\frac{\lambda }{{(n + 2)(n + 4)}}$ to be a constant for simplicity which otherwise will lead to a higher order polynomial expression for k⁺. Even with a higher order polynomial solution for k⁺ the corresponding polynomial coefficients have to be fitted empirically through measurements that will simply make the computations laborious. Hence it is better to assume $\frac{\lambda }{{(n + 2)(n + 4)}}$ to be a single constant.

Therefore Eq. (A5) finally takes the form.

$$k^{ + } = By^{{ + 2}} e^{{ - \left( {\frac{{y^{ + } }}{{A_{3} ^{ + } }}} \right)}}$$

(A6)

where B = $\frac{\lambda }{{(n + 2)(n + 4)}}$ = 0.17 and $\alpha (\text{Re} _{\tau } ) = 0.4147\text{Re} _{\tau } ^{{ - 0.11}}$.

Appendix 2

Equation (11a) is

$$i.e,\;\;\nu _{t}^{{}} = C_{{\nu t}} v^{'} {\text{ }}l$$

(A7)

Approximating $v^{'} \sim \sqrt k$ and using normalised variables ${\text{ }}k^{ + } = k/u_{*}^{2}$, ${\text{ }}\nu _{{\text{t}}}^{ + } = \nu _{t} /\nu$,${\text{ }}l^{ + } = lu_{*} /\nu$, Eq. (A7) can be rewritten as

$$\nu _{t}^{ + } = {\text{C}}_{{\nu {\text{t}}}} {\text{ }}\sqrt {k^{ + } } {\text{ }}l^{ + }$$

(A8)

Employing the near wall van driest damping function $l^{ + } = \kappa y^{ + } \left( {1 - e^{{ - \frac{y}{{A_{d} ^{ + } }}^{ + } }} } \right)$(Pope [1]), Eq. (A8) becomes

$$\nu _{t}^{ + } = {\text{C}}_{{\nu {\text{t}}}} {\text{ }}\kappa \sqrt {k^{ + } } {\text{ }}y^{ + } \left( {1 - e^{{ - \frac{y}{{A_{d} ^{ + } }}^{ + } }} } \right)$$

(A9)

$$i.e.,\;\;\;\nu _{t}^{ + } = {\text{C}}_{{\nu {\text{t}}}} {\text{ }}\kappa {\text{ }}(b^{ + } y^{ + } ){\text{ }}y^{ + } \left( {1 - e^{{ - \frac{{y^{ + } }}{{A_{d} ^{ + } }}}} } \right),$$

(A10)

${\text{where }}\sqrt {{\text{k}}^{ + } } = b^{ + } y^{ + } {\text{,}}$ in the leading order near the wall, Pope [1].

${\text{Writing, }}b^{ + } y^{ + } \sim \left( {1 - e^{{ - \frac{{y^{ + } }}{{A_{d} ^{ + } }}}} } \right){\text{ with }}b^{ + } = \frac{1}{{A_{d} ^{ + } }} < < 1,$ Eq. (A10) becomes

$$\nu _{t}^{ + } = {\text{C}}_{{\nu {\text{t}}}} {\text{ }}\kappa {\text{ }}y^{ + } \left( {1 - e^{{ - \frac{{y^{ + } }}{{A_{d} ^{ + } }}}} } \right)^{2}$$

(A11)