Wind and boundary layers in Rayleigh-Bénard convection. II. Boundary layer character and scaling

Wind and boundary layers in Rayleigh-Bénard convection. II. Boundary layer character and scaling

Abstract

The scaling of the kinematic boundary layer thickness and the friction factor at the top- and bottom-wall of Rayleigh-Bénard convection is studied by Direct Numerical Simulation (DNS). By a detailed analysis of the friction factor, a new parameterisation for and is proposed. The simulations were made of an aspect-ratio domain with periodic lateral boundary conditions at and . A continuous spectrum, as well as significant forcing due to Reynolds stresses clearly indicates a turbulent character of the boundary layer, while viscous effects cannot be neglected judging from the scaling of classical integral boundary layer parameters with Reynolds number. Using a conceptual wind model, we find that the friction factor should scale proportional to the thermal boundary layer thickness as , while the kinetic boundary layer thickness scales inversely proportional to the thermal boundary layer thickness and wind Reynolds number . The predicted trends for and are in agreement with DNS results.

Rayleigh-Bénard convection, wind, DNS, boundary layer, friction factor, turbulence
pacs:
44.25.+f, 47.27.ek, 47.27.eb, 47.27.te

I Introduction

The structure of the boundary layer is of great importance for understanding the turbulent heat transfer characteristics of Rayleigh-Bénard convection. Inherently unstable due to buoyancy effects, the thermal boundary layer with thickness is in a dynamic equilibrium of heating (cooling) by thermal diffusion and the detrainment (entrainment) of heat due to impinging and ejecting thermals at the bottom (top) plate. This process creates large temperature gradients across the boundary layer, thereby enhancing the heat transfer through the wall and thus the Nusselt number . Next to a thermal boundary layer, one can identify a kinematic boundary layer with thickness , associated with the velocity field. Depending on the Prandtl number , which is the ratio between the kinematic viscosity and thermal diffusivity , the kinematic boundary layer can be nested inside the thermal boundary layer or vice versa, which influences the effectiveness of the heat transfer as a function of the Rayleigh number . The Rayleigh number is defined as , where is the thermal expansion coefficient, the gravitational constant, the temperature difference between the top and bottom plate and the domain height. The scaling of and as a function of and are therefore of importance for proper prediction of the heat transfer.

In the theory of Grossmann and Lohse (Grossmann and Lohse, 2000), the wind velocity and the boundary layer thicknesses and are central parameters, which are used to estimate the dissipation rates of kinematic energy and temperature variance in the bulk and the boundary layers. In the theory, and are defined as

(1)
(2)

While (1) holds excellently, the correspondence of (2) with experiments (Xin et al., 1996; Xin and Xia, 1997) and simulations (Kerr, 1996; Kerr and Herring, 2000) is less satisfactory. Relation (2) can be obtained by non-dimensionalising the steady laminar two-dimensional Prandtl boundary layer equations (Schlichting and Gersten, 2000; Grossmann and Lohse, 2004), by which (2) follows immediately. However, the measured dependence of is much weaker than predicted by (2) (see also Fig. 1). It has been suggested that the difference is due to geometry effects Grossmann and Lohse (2003) (plate-filling vs. laterally restricted flow).

In this paper, we argue that the disparity in the expected and the observed scaling is because the top- and bottom boundary layers are not laminar, i.e. forcing due to Reynolds stresses cannot be neglected in the kinematic boundary layer. Consequently, the arguments leading to (2) do not hold. With a detailed DNS study of the momentum- and heat-budgets and the friction factor, and using the wind model of the accompanying paper van Reeuwijk et al. (2007), we derive new parameterisations for and .

A related question is whether or not the boundary layers can be regarded as turbulent. The Reynolds number is too low to sustain a ”classical” turbulent boundary layer ( at ), i.e. a boundary layer where the turbulence production due to shear is in local equilibrium with dissipation. Hence, the general view is that the boundary layers are laminar, but time-dependent. Although time-dependence due to plume impingement and detachment prevents laminarity in the strict sense, the assumption could be justified if the plumes are passive with respect to the scaling of integral boundary layer parameters such as the friction factor and the kinematic boundary layer thickness . Several other studies show that the friction factor scales similar to a Blasius boundary layer (Chavanne et al., 1997, 2001; Amati et al., 2005). However, the scaling of does not comply with classical laminar scaling (2), as discussed before. Furthermore, a recent study of time-spectra in the bottom kinematic boundary layer revealed that the spectra in the boundary layer and in the bulk were practically indistinguishable Verdoold et al. (2008), a strong indication of turbulence. In order to understand this dual behavior, we study several turbulence indicators for the boundary layers, such as the spectra, shape- and friction factor.

The paper is outlined as follows. A brief summary of the code for direct simulation and symmetry-accounted ensemble-averaging is given in section II. The scaling of the boundary layer thickness, the velocity profile, the friction factor and the shape factor are studied in sections III.1, III.2 and III.3 respectively. Then, we study the space and time spectra (section III.4). In section III.5, the mean momentum and temperature budgets in the boundary layers are studied to clarify the importance of fluctuations in the boundary layers. Using the results from the momentum budgets, the friction factor is decomposed in a pressure and momentum-flux contribution in section III.6. This leads to the insight that the main contribution is by the pressure gradient. Using the conceptual wind model derived in the accompanying paper van Reeuwijk et al. (2007), scaling laws for and are derived in section IV. As the results show that the flow has many typical features of turbulence but also of laminarity, the interpretation of the results is discussed in section V. Conclusions are drawn in section VI.

Ii Simulations

Direct simulation of Rayleigh-Bénard convection has been performed at and in a aspect-ratio domain. The code is based on a second-order variance-preserving finite-difference discretisation of the three-dimensional Navier-Stokes equations and is fully parallellised. For all simulations, a grid was used sufficient resolution to resolve the smallest turbulent scales, i.e. the Kolmogorov scale and Corrsin scale . The top and bottom wall are rigid (no-slip) and of fixed temperature. At the side domain boundaries, periodic boundary conditions are applied. For each except the highest, 400 independent realisations were obtained by performing 10 independent simulations and sampling the velocity and temperature field roughly twice every convective turnover time. Because of a formidable computational requirements for , we use this simulation only for the results of Fig. 1 and confine the wind-decomposed analysis to the lower cases, though without loss of generality.

Similar to domains confined by sidewalls, a wind structure develops also in domains with lateral periodic boundary conditions. However, here the wind structure can be located anywhere in the domain since it is not kept in place by sidewalls. To extract the wind, symmetry-accounted ensemble-averaging is used (van Reeuwijk et al., 2005), which aligns the wind structure in different realizations before averaging. In this way a wind structure can be identified unambiguously for these domains, by which a decomposition in wind and fluctuations becomes possible. The resulting average velocity and temperature (three-dimensional fields) are denoted respectively by and . The tildes are used to distinguish the conditional average from the standard (long-time, ensemble or plane) average which is a function of only. The symmetry-accounted average can be interpreted exactly as classical Reynolds-averaged results. For further details we refer to the accompanying paper (van Reeuwijk et al., 2007).

Iii Results

iii.1 Boundary layer thickness

The thickness of the hydrodynamic and thermal boundary layers as a function of is shown in Fig. 1. Here, and are defined as the location of the maximum of mean squared horizontal velocity fluctuations and mean squared temperature fluctuations , respectively. The approximate powerlaws are and respectively, in good agreement with other simulations (Kerr, 1996) and reasonable agreement with experiments (Xin et al., 1996) (despite differences in aspect-ratio, geometry and boundary conditions).

Figure 1: The thermal and kinematic boundary layer thickness and as a function of . The dashed and dash-dotted lines in the graphs are predictions by the G-L theory of and , respectively.

Also shown in Fig. 1 are the predictions of the boundary layer thickness (1), (2) from the Grossmann-Lohse theory (Grossmann and Lohse, 2000), together with the DNS results. The thermal boundary layer thickness is in good agreement with the simulations. The width of the kinematic boundary layer does not agree so well with the G-L theory, as the actual exponent is instead of (where we have assumed free-fall scaling for simplicity).

Below we briefly recapitulate the arguments of Grossmann and Lohse (2004) leading to (2). The starting point is the laminar two-dimensional Prandtl boundary layer equation White (1991); Schlichting and Gersten (2000)

(3)

Upon substituting , , and , the equations become parameter independent as

(4)

Neither this expression, nor the incompressibility conditions, nor the boundary conditions have an explicit dependence on , so the solution has to be independent of as well. Therefore, the flow pattern undergoes a similarity transformation, and the boundary layer thickness scales as . This result is rigorous, provided that (3) holds, i.e. that turbulent stresses do not play a role in the momentum budget. In section III.5 we show that forcing due to Reynolds-stresses cannot be neglected for the boundary layer equations so that the laminarity assumption does not hold.

Figure 2: The horizontal boundary layer velocity profile based on for various . a) semi-log plot and non-dimensionalized with friction velocity . Dashed line represents ; b) normalized by the maximum velocity and the kinematic boundary layer thickness .

iii.2 Velocity profiles

The characteristic shape of the velocity profile can be obtained from the plane-averaged horizontal average velocity as . Figure 2a shows these profiles for various in plus units, i.e. scaled by the friction velocity with and . Here, we define a typical wall shear-stress as

(5)

In Fig. 2a, the viscous sublayer relation is shown with a dashed line, and logarithmic scaling of the velocity profile results in a straight line. For a classical turbulent channel flow and constant-pressure boundary layer, the viscous region ends at , the log-layer starts from and the profiles will collapse onto a single universal curve for all . Here the situation is completely different. First, in plus-coordinates the profiles do not collapse onto a single curve. Furthermore, the viscous region ends at approximately , and the velocity reaches its maximum at at . A region where the velocity scales logarithmically is hard to distinguish, indicating the absence of an inner (constant stress) layer.

Shown in Fig. 2b is the velocity profile normalized by the outer variables, i.e. the boundary layer thickness and the maximum velocity . Although the profiles show that there is a dependence, it is very weak. The weak influence of the number - especially for the two lower numbers considered - is further evidence that the kinematic boundary layer does not behave as a classical forced turbulent boundary layer. Note that the approximate universality of the velocity profiles means that inner and outer variables can be interchanged, in the sense that .

Several experiments have shown universality in upon an outer scaling by boundary layer thickness and maximum velocity (Xin et al., 1996; Lam et al., 2002; Qiu and Xia, 1998), so it is quite interesting that the boundary layer profile found here (Fig. 2b) has a (small) dependence. There may be several reasons for this difference. The experiments have been carried out at much higher , in the range and at higher (the working fluid was water). Furthermore, the presence of side walls and the smaller aspect ratio will be of influence.

It is useful to express the shear-Reynolds number in terms of , and the non-dimensional velocity gradient at the wall. Let the outer scaled variables be denoted by and . The non-dimensional velocity gradient at the wall is connected to the wall-shear stress by , where is the non-dimensional velocity gradient at the wall. Hence, the shear Reynolds number can be expressed as

(6)

All three terms , and depend on , although the dependence of the last term is very weak as .

26 1.02 2.37
52 0.51 2.35
119 0.23 2.27
Table 1: Characteristic numbers for the boundary layer profile at various : the shear-Reynolds number , the shape factor and the friction coefficient .

iii.3 Friction and shape factor

The friction and shape factor (e.g. Schlichting and Gersten, 2000; White, 1991) have been calculated for all three (Table 1). The friction factor is defined as

(7)

Here we note that combining (6), (7) and neglecting the small dependence of the wall-gradient gives that can be approximated by

(8)

This is consistent with the approximation , which is an important assumption in the Grossmann-Lohse theory Grossmann and Lohse (2000). The observation (8) will prove to be important to establish the scaling of in section IV.

Based on the values of Table 1 and in terms of , the friction factor scales as . An empirical relation for turbulent plane channel flow is , with based on channel half width and mean velocity across the channel (Dean, 1978). The friction factor of laminar boundary layers have a stronger dependence on ; for plane Poiseuille flow ( based on full channel height) and for the Blasius flat plate flow . Hence, judging from the scaling of friction factor, the behavior of the boundary layer would be classified as laminar. These results are consistent with Chavanne et al. (1997, 2001); Amati et al. (2005).

Figure 3: -averaged wind structure at and . Colorscheme is by the relative temperature . Dark areas are relatively cold, and white areas are relatively hot.

The shape-factor is defined as , where and are the displacement and momentum thickness, given by:

For laminar profiles, such as Poiseuille flow and the Blasius solution for the developing flow over a flat plate, the shape factor is approximately (e.g. Schlichting and Gersten, 2000; White, 1991). For turbulent plane channel flow, flat-plate constant-pressure boundary layers and a plane turbulent wall jet (Rajaratnam, 1976) the shape factor is approximately . Based on this information, the values from Table 1 indicate that the velocity profile follows a laminar-like distribution with a slight trend towards turbulent values as increases.

If the shape and friction factor are taken to be representative to distinguish a laminar from a turbulent boundary layer, the boundary layer would be classified as laminar. In the next sections we will study the momentum budgets of the boundary layers, and compare the time and space spectra of boundary layer and the bulk. It will be shown that from this perspective, the kinematic boundary layer has many features of turbulence.

iii.4 Fluctuations and spectra

Figure 4: Close-up of horizontal velocity and turbulence intensities (legend in Fig. 4b) at the position with the maximum horizontal velocity. (a) ; (b) ; (c) . The horizontal dashed line indicates the edge of the thermal boundary layer .

In Fig. 4a-c the average velocity profile is shown for and , together with the turbulence intensity of the horizontal and vertical fluctuations, and respectively. These are the profiles of the -averaged wind structure (see Fig. 3), with the -location chosen such that the horizontal velocity is at its maximum, i.e. where the flow is parallel to the wall and from left to right. A striking feature of the turbulence intensity of the horizontal fluctuations, is that it is so large compared to the mean wind, namely 70-80%. For turbulent channel flow, typical turbulence intensities are 5-10%. Outside the thermal boundary layer the horizontal turbulence intensity is constant. The vertical turbulence intensity is not as large as the horizontal due to wall blocking, but is still 20% at the edge of the thermal boundary layer, and 50% at the edge of the kinematic boundary layer. This confirms that fluctuations in large aspect-ratio domains are larger relative to the wind Niemela and Sreenivasan (2006), in comparison with small aspect-ratio domains (e.g. (Xin et al., 1996) reports turbulence intensities of 20%).

One of the main features of turbulence is the presence of a continuous range of active scales. A simulation at is used to obtain both spatial and temporal spectra of the horizontal velocity components. To collect temporal spectra, eight points have been monitored: four bulk and four boundary layer points. The bulk points are taken at and the boundary layer points were chosen according to . The temporal spectra are generated by segmenting the time series and a Welch window has been used. Then, averaging was performed over the spectra of the two horizontal velocity components and the four monitoring points. The spatial spectra were collected by performing a 2D FFT and integrating over circles and averaging over approximately 10 turnovers.

Figure 5: Spectra of the horizontal velocity components in the boundary layer and in the bulk at normalized by the bulk variance. (a) temporal spectrum (b) spatial spectrum.

The temporal spectra of the horizontal velocity components at are shown in Fig. 5a. There is a continuous range of active scales which spans about two decades, although turbulence production and dissipation are not sufficiently separated to form a clearly discernible inertial subrange. The spatial spectra (Fig. 5b) also reveal a continuous range of active scales.

What is striking about the spectra of the bulk and the boundary layer is how similar they are, both in range of active scales and in amplitude. Despite a mild damping at the intermediate frequencies and wave numbers, the similarity indicates that the dynamics of the bulk and the boundary layer - both temporal and spatial - are very similar. We note that the simulation at is well inside the hard-turbulence regime. The transition to hard turbulence occurs at much lower for large aspect-ratio domains than the generally accepted value of (Heslot et al., 1987). Indeed, for aspect-ratio 6 domains, the flow has hard-turbulence regime scaling occurs from upwards (Kerr, 1996). If one accepts that the flow core is turbulent, then Fig. 5 indicates that the boundary layers are turbulent as well.

The striking similarity between the spectra in the bulk and the boundary layers seems to be a robust and general feature of Rayleigh-Bénard convection. In a recent paper (Verdoold et al., 2008), we present combined experimental and numerical results of an aspect-ratio 4 cavity filled with water for Rayleigh numbers ranging from to . For all from upwards, it is found that the spectra in the bulk and the boundary layer are practically identical.

iii.5 Momentum budgets

(a) budget -
(b) budget -
(c) budget -
(d) budget -
(e) budget -
(f) budget -
(g) budget -
(h) budget -
(i) budget -
Figure 6: Momentum- and heat-budget in the boundary layer at the position with the maximum horizontal velocity for various : (a,d,g): ; (b,e,h): ; (c,f,i): .. (a,b,c); -momentum budget; (d,e,f): -momentum budget; (g,h,i): -budget. The legend for Figs. (a-i) is shown in Fig. ie and the horizontal dashed line indicates the edge of the thermal boundary layer .

Momentum budgets are a very direct way to get an impression of the importance of the turbulent Reynolds stresses. As before, -averaged results (Fig. 3) are used for convenience of presentation. Checks have been made to ensure that the budgets shown here are also representative for the three-dimensional field. The -location has been chosen such that the horizontal velocity is at its maximum, i.e. where the flow is parallel to the wall and from left to right. This guarantees that horizontal gradients are small, and that no adverse of favorable pressure gradients are present. Shown are budgets for (Fig ia,d,g), (Fig ib,e,h) and (Fig ic,f,i). The budgets for the horizontal (Figs. ia-c) and vertical momentum (Figs. id-f) have been nondimensionalized by , and heat budget (ig-i) by . The legend for the budgets is shown in Fig. ie and the budget terms are defined in Table 2. The -coordinate has been scaled by and the horizontal dashed line denotes . For reference, the ratio is 0.8, 0.6 and 0.38 for the simulations at , and respectively.

Table 2: Budget terms for momentum and heat equation.

For the horizontal momentum budgets (Figs. ia-c), the balance is between the horizontal pressure gradient and diffusion for . Outside the thermal boundary layer, is not negligible; on the contrary, fully balances the pressure gradient near . This indicates that the turbulence outside the thermal boundary layer is key to the boundary layer thickness, as will be outlined in section IV. As the location of the budgets has been chosen such that all horizontal derivatives are small, and .

Log-scaling is expected in the inner layer where is constant, so that . For channel flow, is zero at the wall and peaks in the buffer layer, marking the transport of momentum from the outer to the inner layer. After the peak, it crosses the zero axis where the log-layer is expected. This behavior of is absent for , but as increases a peak forms inside the thermal boundary layer (Fig. ia-c). However, in terms of forcing is always much larger than the small peak for the range of under consideration, which again confirms that this is not a classical forced turbulent boundary layer.

Figs. id-f show the budgets of the -momentum equation. Here the balance is between buoyancy , the vertical pressure gradient and the Reynolds stresses . Very near the wall, roughly in the lower half of the thermal boundary layer, the buoyancy and pressure are in balance, so the flow is neutrally buoyant here. Further away from the wall, at the edge of the thermal boundary layer, the contribution of is significant, even if it may seem small compared to the near-wall (hydrostatic) balance of and . In fact, comparing of the vertical momentum equation to the magnitude of terms in the horizontal momentum equation shows that it is of the same magnitude as . Outside the boundary layer, the pressure gradient is positive and is balanced purely by fluctuations .

The -momentum budgets (Figs. ih-i) show a balance between thermal diffusion , turbulence and there is a contribution from advection . Judging from the peak of around , the nonzero contribution of to the heat budget is probably caused by some spatial variations in by which . The peak of and is always located just inside the thermal boundary layer, representing the location where diffusion and fluctuations most effectively exchange heat.

It is striking that the dominant length scale for the budgets is the thermal boundary layer thickness (which is denoted by the horizontal dashed line in Fig. i), and not as one may expect, the kinematic boundary layer thickness. Perhaps this should not be too much of a surprise, as the thermal boundary layer thickness can be well represented by , and the Nusselt number represents the efficiency of the convective heat transfer mechanism of the flow, resulting from the non-linear coupling of temperature and velocity under the action of buoyancy. Therefore, is equally important for the heat-budget and for the momentum budgets. In fact, is a dominant parameter in the scaling of both and , as will be shown in section IV.

The findings of Figs. ia-i can be summarized as follows for the , and budget, respectively:

(9)
(10)
(11)

These equations represent the boundary layer equations at the -location where the flow is parallel to the wall and horizontal derivatives are negligible (roughly halfway between the impingement and detachment region). Note that even though the -momentum equation is not directly coupled to the other two equations, the vertical fluctuations are non-trivially coupled to and as these terms represent to a large extent the plumes emerging from and impinging on the boundary layers. The equations above are two-dimensional, but by the absence of transversal derivatives, it can be expected that these equations are valid for the three-dimensional case as well, in a local coordinate system aligned with the flow and at the location where the flow is parallel to the wall.

The boundary layer equation (9) clearly shows that one cannot neglect the influence of turbulence in the boundary layer dynamics. Hence, the laminar boundary layer equation (3), which lead to the scaling is not valid: additional information is required about to estimate . In section IV, the scaling behavior of will be derived using flow-specific information obtained from the DNS results.

iii.6 The friction factor decomposed

By using the boundary layer equation (9), the dominant contributor to the friction factor can be identified. Integrating (9) over the kinematic boundary layer and substituting (7), the friction factor is composed of a contribution from pressure and a turbulent momentum flux as

(12)

The terms on the right hand side of (12) have been calculated with the DNS results and are presented in Table 3 1 The decomposition clearly demonstrates that is dominated by the pressure gradient. The turbulent momentum flux is small but positive, i.e. a flux out of the boundary layer. Hence, we conclude that the dynamics of the wall-friction is not governed by turbulence (i.e. Reynolds stress) as in a forced turbulent boundary layer. In the latter, the dynamics is dominated by a large momentum-flux into the boundary layer, while the contribution of the pressure gradient is negligible.

. 0.81 0.88 -0.07 0.39 0.42 -0.03 0.17 0.19 -0.02

Table 3: Decomposition of the friction factor according to (12)
Figure 7: The effective forcing , which vanishes quickly outside the thermal boundary layer.

Using (12), can be parameterised. Shown in Fig. 7 is the effective forcing . To first order, for , while for the Reynolds stress forcing balances the pressure to that . Hence, can be estimated by

(13)

Clearly, (13) holds at moderate only, when turbulent shear production in the boundary layer is small. The formation of the peak inside the thermal boundary layer at and (Fig. 7) suggests that shear production becomes more important as increases, and this will have to be accounted for in (13) at higher . However, it was shown in the accompanying paper van Reeuwijk et al. (2007) that the wind velocity becomes independent of at sufficiently high , because is negligible compared to the mixing parameter . Therefore, incorrect scaling behavior in (13) will not influence the wind dynamics at high .

Iv Scaling of and

Using the simple two-equation wind model derived in the accompanying paper van Reeuwijk et al. (2007), we can establish the scaling behavior of and . The model uses a dimensionless wind-velocity and spatial temperature difference , where is the free-fall velocity. The governing equations of the model are given by

(14)
(15)

Here where is the typical roll size, , and . The turbulent Prandtl number and the mixing parameter are coefficients with values Schlichting and Gersten (2000) and respectively. The pressure difference which drives the wind is generated by a spatial temperature difference (it is relatively hot where the flow ascends and relatively cold where it descends, see Fig. 3). The temperature difference is in its turn generated by large horizontal heat fluxes originating from the interaction between the mean wind and temperature field. The model depends on , and , where and have to be provided. Based on the analysis of the friction factor (section III.6), an explicit expression for can be derived, by which the model only depends on empirical input for (and thus ).

The steady state estimate for the pressure gradient at the bottom wall of the wind model is van Reeuwijk et al. (2007)

(16)

Using (16), (13) can be further specified as

(17)

Hence, the wall friction term is linear in the temperature difference as

(18)

Here, we assumed that . With (18), the empirical specification of is no longer necessary, and the model is given by:

(19)
(20)
Figure 8: The predictions of the wind model (eqns (19), (20), thick lines) compared to the DNS results (diamonds) for a) the typical wind ; b) the spatial temperature difference ; c) the friction factor and d) kinematic boundary layer thickness .

The steady state solution of the model as a function of is shown in Fig. 8. At this point, the only empirical data used in the model is the powerlaw for and the roll size . The mixing parameter is kept at the same value as in van Reeuwijk et al. (2007), namely 0.6. As can be seen, the model captures the trends of , and satisfactorily. Note that the profiles could be made to match quantitatively as well when one would introduce some additional coefficients. However, the focus of this paper is not to develop a carefully tuned model, but to elicit general scaling behavior.

It is not very useful to have an expression for in terms of , as this quantity is rarely measured. However, by using the steady state solution of (19), can be expressed in terms of as

(21)

so that the dependence of can be eliminated. Using (18) and (21), is given by

(22)

Hence, when , the model predicts that . Note that the term in the denominator represents the effects of wall friction. Hence, when , scales independently of wall-effects. It is the turbulence in the outer flow which fully determines the velocity at the edge of the boundary layer.

A scaling relation for can be derived by using the two different expressions for , (8) and (17). This results in

Using , is approximated by

(23)

Dropping the absolute signs and using (21), is given by

(24)

Upon assuming that , it follows that scales as . Fig. 8d shows the prediction of the wind model for . Although the boundary layer thickness is underpredicted, the trend is in agreement with the DNS data. Given the simplicity (with only one calibration parameter ), the model captures the trends of wind velocity, spatial temperature difference, friction factor and kinematic boundary layer thickness satisfactorily.

V Turbulent or not?

The apparently contradicting findings reported in the previous sections is quite intriguing. On the one hand, the results indicate that the kinematic boundary layer is turbulent. The deduced boundary layer equation (9) shows that forcing due to turbulent Reynolds stresses is significant, in particular outside the thermal boundary layer. Furthermore, the spectra in the bulk and the boundary layers are nearly indistinguishable and show the existence of a continuous range of active scales both in space and time. Both are an indication for turbulence.

On the other hand, the results suggest that the kinematic boundary layer does not correspond to a classical turbulent boundary layer. The Reynolds numbers in the range we consider ( at ) are generally considered too low to sustain turbulence. Moreover, the friction factor for a classical forced boundary layer has a weak dependence on (reflecting the quadratic wall friction), and is dominated by the turbulent momentum-flux from the free stream. For the boundary layers under consideration, the dominant contributor to is the pressure gradient (section III.6) and not the momentum-flux. Consequently, has a significant dependence. The near-universal profiles (found in the present work especially for the two lower numbers) as a function of based on the outer variables and (Fig. 2) are further evidence against a classical turbulent boundary layer: a turbulent boundary layer can by definition not be universally scaled by outer variables.

The difference between classical forced turbulence boundary layers and a boundary layer of Rayleigh-Bénard convection may be best characterized by the way the turbulence is produced and redistributed. For forced flow cases, turbulence cannot be maintained at low , as the dissipation in the boundary layer will be stronger than the production. However, for Rayleigh-Bénard convection the production and transport of turbulent kinetic energy (TKE) is not confined to the inner layer alone. Instead, TKE is produced in the bulk, where it is partially dissipated. The surplus is transported to the boundary layer by pressure velocity fluctuations (see also Kerr, 2001). Therefore, there is no local equilibrium between production and dissipation, and turbulence can be maintained in the boundary layers below the critical . At sufficiently high , instabilities due to shear can be expected to maintain themselves, and several experiments and simulations show such a transition around (Chavanne et al., 2001; Niemela and Sreenivasan, 2003; Amati et al., 2005).

A simple explanation for the laminar-like scaling of classical integral boundary layer parameters may be that the forcing in the wall-parallel direction is very weak compared to the forcing in the wall-normal direction (plume impingement and detachment). Indeed, the forcing in the vertical direction is the direct result of buoyancy, while the pressure gradient in the horizontal direction forms is due to large-scale differences in mean temperature. This can be made explicit by considering the ratio of forcing in the wall-normal direction (buoyancy) and wall-parallel direction (16), which is given by

(25)

At , this ratio is approximately 50, and at , the ratio is approximately 100. Thus, the boundary layers under consideration here are forced primarily in the wall-normal direction, and the force generating the wind is relatively weak.

Despite the laminar-like scaling of the integral parameters, a parallel can be drawn with a fully developed forced boundary layer: both have a viscous sublayer dominated by viscosity which suppresses instabilities and prevents their growth and development of turbulence. However, as demonstrated by seminal experiments in the sixties Kline et al. (1967), despite linear velocity variation, the flow within the sublayer in a forced boundary layer is not laminar, but accompanied by considerable irregular fluctuations, streaks and other structures. One can argue that the same dynamics occur in the boundary layers of Rayleigh-Bénard convection. In particular, Figs. ia-c indicate that the thermal boundary layer functions as a viscous sublayer, and the region as an cross-over region between the exterior flow and the thermal boundary layer. The absence of a constant stress layer dominated by the turbulent momentum flux suppresses a logarithmic region and marks a fundamental difference with forced turbulent boundary layers.

Vi Conclusions

The aim of this paper has been to study the boundary layers which develop under the joint action of plumes and wind in Rayleigh-Bénard convection at the top and bottom plates. Direct numerical simulation was used for simulations at and for aspect ratio domains with periodic side boundary conditions. For each , 10 independent simulations have been carried out, resulting in approximately 400 independent realizations per . Processing the results using symmetry-accounting ensemble averaging made it possible to retain the wind structure, which would normally cancel out due to the translational invariance of the system.

The importance of Reynolds-stresses in the boundary layers, as well as the temporal and spatial spectra indicate undoubtedly a turbulent character of the boundary layer. However, the behavior is rather different from classical forced boundary layers, as can be judged from the laminar-like scaling of the classical integral boundary layer parameters. Indeed, viscous effects play an important role within the thermal boundary layer, and a large turbulent momentum-flux from the external stream is absent. This difference is probably caused by the fact that the turbulence inside the kinematic boundary layer of RB originates from the bulk, whereas classical forced boundary layers are in a local equilibrium between production and dissipation of turbulent kinetic energy.

Due to the importance of Reynolds stresses in the boundary layer, the arguments underpinning the kinematic boundary layer scaling do not hold. Using the DNS results and a conceptual wind model van Reeuwijk et al. (2007), explicit expressions for and were derived. It was found that the friction factor should scale proportional to the thermal boundary layer thickness as . The kinematic boundary layer thickness scales inversely proportional to the thermal boundary layer thickness and the Reynolds number as . The predicted trends for and are in agreement with the DNS results.

With the closure for , the model (19), (20) depends solely on empirical input for , and predicts the wind Reynolds number , friction factor and kinematic boundary layer thickness .

Acknowledgements.
This work is part of the research programme of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). The computations were sponsored by the Stichting Nationale Computerfaciliteiten (NCF).

Footnotes

  1. The values for in Table 3 are slightly smaller than those in Table 1, as the latter is deduced from the plane-averaged squared mean velocity.

References

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