The near wall physics and wall functions for turbulent natural convection

doi:10.1016/j.ijheatmasstransfer.2011.12.031

International Journal of Heat and Mass Transfer

Volume 55, Issues 9–10, April 2012, Pages 2625-2635

https://doi.org/10.1016/j.ijheatmasstransfer.2011.12.031 Get rights and content

Abstract

Highly accurate results of direct numerical simulations (DNS) for Grashof numbers up to 4.0 × 10⁶ in a differentially heated infinite vertical channel are used to deduce wall functions for turbulent natural convection. These functions represent the unique time-averaged behaviour of velocity, temperature, and shear stress in the vicinity of the wall for the Grashof number range under consideration. There is a good indication that these wall function are valid as the Grashof number tends to infinity. Previous attempts to find such wall functions relied on the blending of at least two functions which are valid in adjacent regions of the flow field. In conformity with the time-averaged momentum and thermal energy equation, this study introduces a continuous description of the near wall region up to the velocity maximum.

Introduction

The near wall region of a turbulent flow field is the most challenging part for turbulence modelling. Especially the physics of the turbulent heat fluxes change rapidly within a very thin layer. Thus, it is very attractive to find a universal distribution of the mean velocity and temperature that can be used as wall functions.

A wall function acts as a new boundary condition, hence, not only the velocity and temperature but also the flow variables of the turbulence model have to be prescribed. The latter is often poorly considered so that Reynolds averaged Navier–Stokes (RANS) computations using wall functions lead to erroneous results due to inadequate boundary conditions for the turbulence model.

The main advantage of RANS computations using wall functions is to avoid a high resolution of the turbulent near wall layer and uncertainties due to inappropriate turbulence models. For numerical implementations, this requires a continuous function for the mean flow values, since for the initial flow field the first grid point away from the wall may lie inside the viscous sublayer. At the end of the iteration, however, it may be located within the fully turbulent regime although the grid was fixed in space. Moreover, in a developing flow the boundary layer thickness smoothly changes which again requires a smooth wall function. Therefore it is not sufficient to have wall functions that are only valid in a certain part of the turbulent near wall region.

In practice, wall functions are identified for several regions and then blended into each other in order to provide this smooth behaviour. For forced convection the law of the wall or log-layer is well established in the literature although it is still the subject of some ongoing discussions (see Barenblatt [1]; Zanoun et al. [14]). The main problem herein is that results of direct numerical simulations (DNS) even nowadays are still not able to clearly show the logarithmic regime that can be seen in experiments, since the highest possible Reynolds numbers are still not high enough. Experiments, on the other hand, suffer from improper measurements at high Reynolds numbers due to the steep gradients close to the wall. This especially refers to measuring the wall shear stress which is a reference quantity in the universal nondimensional wall functions (see Fernholz et al. [4]).

With these challenges is mind, new wall functions for pure natural convection are derived. They are based on DNS results in a differentially heated vertical channel with the intention to generally use them in natural flow situations. These wall functions will smoothly cover the whole near wall region even beyond the maximum of the mean velocity profile. Thus, common problems when properly predicting the production of turbulent kinetic energy in natural convection using eddy viscosity models, can be avoided [3]. Therefore, not only wall functions for temperature and velocity are presented but also for the turbulent shear stress $\bar{u^{'} v^{'}}$ in order to define new boundary conditions for turbulence modelling. Due to several reference quantities used in the course of this paper it must be highlighted that indexing is crucial. An upper ∗ denotes a dimensional quantity such as the temperature T^∗ in Kelvin while no upper index refers to a straight forward nondimensionalisation, e.g. using the temperature difference ΔT^∗ as a reference temperature or the channel half width δ^∗ as a reference length. The indices + and × denote special nondimensionalisations that will be used for wall functions.

Section snippets

The vertical channel as a benchmark scenario for natural convection

The following analysis and all figures are based on our new results of DNS computations for the differentially heated vertical channel shown in Fig. 1. For a broad and consistent data set we performed all computations by means of a pseudo-spectral flow solver described and validated in [7]. These results, including turbulent budgets, are highly accurate and obtained for Grashof numbers ranging from the laminar flow regime up to the fully turbulent regime with Gr = 4.0 × 10⁶ as the highest value.

A critical assessment of wall functions in natural convection

Some proposals for wall functions in natural convection made in the past will be discussed in the following. A power law for the temperature profile based on a two layer structure of the turbulent boundary layer was suggested in Versteegh and Nieuwstadt [13]. Their scaling arguments were similar to those of the early work by George and Capp [5]. Later and in contrast to [5], Hölling and Herwig [6] identified one single reference temperature as characteristic for the heat transfer problem as a

A new temperature wall function

Time-averaged temperature DNS data for increasing Grashof numbers in the vertical channel are shown in Fig. 7.

It turns out that the A-values, defined as A ≡ y⁺∂θ⁺/∂y⁺, for increasing Grashof numbers for a wide range of y⁺-values almost perfectly approach a probability density functional (pdf) form. This is shown in Fig. 7(b), where $A \equiv y^{+} \frac{\partial θ^{+}}{\partial y^{+}} = α \exp [- β^{2} (\ln y^{+} - δ)^{2}] + γ$ is specified with respect to the four constants α, β, γ, and δ. These constants are determined so that the maximum of A at y⁺ = 1.095 is

A new velocity wall function

The major advantage of the temperature wall function θ⁺(y⁺) derived in the previous section is its continuous analytical representation of θ⁺. Now a corresponding velocity wall function, consistent with the time averaged momentum equation, will be deduced. From the mean momentum equation, cf. Eq. (1a), (1b), (1c), $0 = - \frac{\partial}{\partial y} \bar{u^{'} v^{'}} + K_{u} \frac{\partial^{2} \bar{u}}{\partial y^{2}} + K_{g} \bar{θ},$ the velocity $\bar{u}$ is found after a formal integration: $\bar{u} = {|\frac{\partial \bar{u}}{\partial y}|}_{w} y - \frac{K_{g}}{K_{u}} \underset{①}{\underset{︸}{\int_{0}^{y} \int_{0}^{y} \bar{θ} dydy}} + \frac{1}{K_{u}} \underset{②}{\underset{︸}{\int_{0}^{y} \bar{u^{'} v^{'}} dy}}$ As a first step, Eq. (17) is used to find a wall function $\bar{u} ($

Wall functions in CFD-codes

The wall functions developed in this study can be implemented in CFD-codes to get an efficient way to calculate turbulent natural convection. They are:

(i)
temperature wall functions θ⁺(y⁺): Eq. (13) with g(y⁺) as in Eq. (14) and the constants in Table 1
(ii)
velocity wall function u^×(y^×): Eq. (30) with F(y⁺) and G(y) as in Eqs. (26), (29), with the constants in Table 1 and the constants a, b, c, d, and y_ip as in Eqs. (28a), (28b), (28c), (28d), (28e)
(iii)
shear stress wall function $\bar{u^{'} v^{'}} (y)$ : (27) and the

Conclusion

As long as Reynolds-averaged equations are used in a theoretical approach to turbulent momentum and heat transfer problems, it is very attractive to have wall functions. They analytically represent the wall nearest part of the flow field which otherwise would have to be calculated with high grid density.

The new wall functions for natural convection are highly accurate smooth functions from the wall up to the velocity maximum including the turbulent shear stress $\bar{u^{'} v^{'}}$ . All these features are

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The near wall physics and wall functions for turbulent natural convection

Abstract

Introduction

Section snippets

The vertical channel as a benchmark scenario for natural convection

A critical assessment of wall functions in natural convection

A new temperature wall function

A new velocity wall function

Wall functions in CFD-codes

Conclusion

Int. J. Heat Fluid Flow

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Scaling laws for fully developed shear flow. Part 1. Basic hypotheses of experimental data

J. Fluid Mech.

Spectral Methods in Fluid Dynamics

New developments and applications of skin-friction measuring techniques

Meas. Sci. Tech.

Asymptotic analysis of the near wall region of turbulent natural convection flows

J. Fluid Mech.