Turbulent natural convection scaling in a vertical channel

doi:10.1016/j.ijheatfluidflow.2013.08.011

International Journal of Heat and Fluid Flow

Volume 44, December 2013, Pages 554-562

https://doi.org/10.1016/j.ijheatfluidflow.2013.08.011 Get rights and content

Highlights

•
DNS mean temperature data for $Ra ⩽ 2.0 \times 10^{7}$ continue to support a −1/3 power-law.
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Implicit Nu–Ra equation is acceptable for $5.4 \times 10^{5} ⩽ Ra ⩽ 2.0 \times 10^{7}$ .
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Inner temperature fluctuations scale with inner temperature and length scale.
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Inner mean velocity and normal velocity variances scale with outer velocity scale, u_o.
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Incipient proportional relationship for u_τ/u_o for $5.4 \times 10^{5} ⩽ Ra ⩽ 2.0 \times 10^{7}$ .

Abstract

Using direct numerical simulation (DNS) data, this study appraises existing scaling laws in literature for turbulent natural convection of air in a differentially heated vertical channel. The present data is validated using past DNS studies, and covers a range of Rayleigh number, Ra between 5.4 × 10⁵ and 2.0 × 10⁷. We then appraise and compare the various scaling laws proposed by Versteegh and Nieuwstadt, 1999, Hölling and Herwig, 2005, Shiri and George, 2008, George and Capp, 1979 with the profiles of the mean temperature defect, mean streamwise velocity, normal velocity fluctuations, temperature fluctuations and Reynolds shear stress. Based on the arguments of an inner (near-wall) and outer (channel centre) region, the data is found to support a minus one-third power law for the mean temperature in an overlap region. Using the inner and outer temperature profiles, an implicit heat transfer equation is obtained and we show that a correction term is non-negligible for the present Ra range when compared with explicit equations found in literature. In addition, we determined that the mean streamwise velocity and normal velocity fluctuations collapse in the inner region when using the outer velocity scale. We also find that the temperature fluctuations scale in inner coordinates, in contrast to the outer scaling behaviour reported in the past. Lastly, we show evidence of an incipient proportional relationship between friction velocity, u_τ, and the outer velocity scale, u_o, with increasing Ra.

Introduction

In the building industry, it is necessary for ventilation-design engineers to understand the control of heating, ventilation and air-conditioning parameters of occupied spaces, because, any form of control influences the design of buildings. One of the prevalent industry practices is to control the heat gain or loss of the building by insulating the external building envelope (Batchelor, 1954). This can take various forms such as double-paned windows, insulated cavity in external walls and double-skin façades. In general, we find that for such forms, the air flow is naturally-convected and wall-bounded. So, to study the insulating nature of these flows, the present study aims to appraise the scaling arguments and wall functions (or wall models) in existing literature and in light of newer, higher-resolution direct numerical simulation (DNS) data.

For computational fluid dynamics (CFD) simulations such as Reynolds-averaged Navier–Stokes simulations (RANS) and large-eddy simulations (LES), it is important to have accurate wall models, especially since these models set the boundary conditions for calculations close to the wall. These models reduce the added computational cost associated with fine grid spacings, a requirement for capturing rapid changes in flow physics near the wall (Kiš and Herwig, 2012). As such, it is important to have accurate wall models that are valid for a large range of flow conditions.

Many studies in the past have developed wall functions using various scaling analyses (e.g. George and Capp, 1979, Yuan et al., 1993, Hölling and Herwig, 2005, Shiri and George, 2008). These wall functions are typically formed with unknown constants, and then fitted to experimental and numerical data (e.g. Versteegh and Nieuwstadt, 1999, Henkes and Hoogendoorn, 1990). However, the opinions of these studies have largely been found to differ, possibly due to the limited numerical data at the time.

This research focuses on validating the aforementioned scaling analyses and wall functions with DNS data for Rayleigh numbers (Ra) up to 2.0 × 10⁷. To date, the DNS data by Versteegh and Nieuwstadt (1999) for Ra up to 5.0 × 10⁶ has been most frequently cited, while the more recent study by Kiš and Herwig (2012) of Ra up to 2.3 × 10⁷ is the highest-known DNS data available for comparison. From the DNS data, Versteegh and Nieuwstadt (1999) proposed constants for their wall functions based on empirical fitting of their theoretical arguments to the data. However, Shiri and George (2008) questions the validity of the existing data at that time for appraising asymptotic theories and argues that, in order for these theories to be evaluated, the data should originate from a flow with a ratio of outer to inner length scales, h/l_i, greater than 10. This criterion is used throughout our simulations, and have been calculated to range between 19 and 62.

Section snippets

Governing equations

In the present study, we adopt the Boussinesq approximation (constant fluid properties except buoyancy, which is a function of temperature) for fully developed turbulent natural convection driven by temperature difference between the two walls, ΔT. Here, ΔT = T_h − T_c, where T_h and T_c are defined as the temperatures of the hot and cold walls respectively. For all simulations in this study, the hot wall is located at the left wall (see Fig. 1) and the reference temperature, T_ref = (T_h − T_c)/2, is defined

Comparison with published DNS data

To validate the present simulation, we compared the statistics with the data of Versteegh and Nieuwstadt, 1999, Pallares et al., 2010 for Ra = 5.4 × 10⁵. For brevity, we report only the turbulent statistics, and this is shown respectively in Fig. 2a and b. Overall, we found that our present data is consistent with published DNS data: the temperature fluctuation profiles in Fig. 2b and turbulent statistics in Fig. 2a matches closely, with the exception of the peak streamwise velocity fluctuations at

Comparison of scaling analyses

We begin by integrating Eq. (2.2), giving $α \frac{d \overline{T}}{dz} - \overline{w^{'} T^{'}} = - \frac{q_{w}}{ρ C_{p}} \equiv f_{w},$ which describes a characteristic heat flux constant, f_w, equivalent to the wall heat flux, q_w, flowing from left to right divided by density, ρ, and specific heat, C_p. From Eq. (4.1), it follows that f_w is independent of location and can be deduced as a characteristic parameter for describing the flow in the channel. We now describe the inner-outer scaling approach—adopted by Versteegh and Nieuwstadt, 1999, Hölling and Herwig, 2005

Temperature profiles

By virtue of adopting the two layer approach described in Section 4, we argue that the outer temperature scale depends on the channel half-width, h. Hence, from Table 2, Table 3, the inner and outer temperature scales are: $T_{i} = {(\frac{| f_{w} |^{3}}{g β α})}^{1 / 4}, T_{o} = {(\frac{| f_{w} |^{2}}{g β h})}^{1 / 3},$ with the respective inner and outer length scales: $l_{i} = {(\frac{α^{3}}{g β | f_{w} |})}^{1 / 4}, and l_{o} = h .$

Using the scalings above and the gradient-matching approach, the temperature wall functions take the following power-law forms: $\frac{T_{h} - \overline{T}}{T_{i}} = - c_{1} {(\frac{z}{l_{i}})}^{- 1 / 3} - c_{2} (\Pr),$ $\frac{\overline{T} - T_{ref}}{T_{o}} = c_{1} {(\frac{z}{l_{o}})}^{- 1 / 3} + c_{3}$

Mean velocity profile

Here, we directly appraise the two inner velocity scales proposed (from Table 2): u_i,h = (gβ∣f_w∣h)^1/3 and u_i,α = (gβ∣f_w∣α)^1/4, by scaling the present data with both velocity scales and plotting against the inner length scale, l_i.

The noticeable difference between the two velocity scales is the choice of the parameter h over α, the latter of which is proposed by Versteegh and Nieuwstadt, 1999, Hölling and Herwig, 2005 based on the classical two-layer approach. Conversely, Shiri and George (2008)

Velocity variances

A cursory analysis of the mean equation of motion (Eq. (2.1)) does not appear to immediately yield any basis for scaling arguments for the velocity variances. However, from the success of collapsing the velocity profiles in Section 6.1, we hypothesise that the velocity variances can be scaled with the velocity scale, $u_{i, h}^{2} = {(g β | f_{w} | h)}^{2 / 3}$ . The scaled profiles of normal velocity variances from this study are shown in Fig. 11a, and compared with the DNS data from Versteegh and Nieuwstadt, 1999, Kiš

Conclusions

From the present DNS dataset for 5.4 × 10⁵ ≲ Ra ≲ 2.0 × 10⁷, we show evidence that provide additional support for a minus one-third power law for the mean inner and outer temperature wall function in an overlap region (see Fig. 4). This is substantiated by the fitting of constants for the wall functions based on compensated temperature gradients Fig. 3. In addition, we show that the resulting heat transfer equation—obtained by evaluating the inner and outer mean temperature wall functions—contains a

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Turbulent natural convection scaling in a vertical channel☆

Highlights

Abstract

Introduction

Section snippets

Governing equations

Comparison with published DNS data

Comparison of scaling analyses

Temperature profiles

Mean velocity profile

Velocity variances

Conclusions

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

J. Comput. Phys.

Int. J. Heat Mass Transfer

J. Comput. Phys.

Int. J. Heat Mass Transfer